Theorem ovolicc2lem4 23507

Description: Lemma for ovolicc2 23509. (Contributed by Mario Carneiro, 14-Jun-2014.) (Revised by AV, 17-Sep-2020.)

Hypotheses

Ref	Expression
ovolicc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
ovolicc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
ovolicc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ovolicc2.4	⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
ovolicc2.5	⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
ovolicc2.6	⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
ovolicc2.7	⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
ovolicc2.8	⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
ovolicc2.9	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
ovolicc2.10	⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
ovolicc2.11	⊢ (𝜑 → 𝐻:𝑇⟶𝑇)
ovolicc2.12	⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
ovolicc2.13	⊢ (𝜑 → 𝐴 ∈ 𝐶)
ovolicc2.14	⊢ (𝜑 → 𝐶 ∈ 𝑇)
ovolicc2.15	⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
ovolicc2.16	⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}
ovolicc2.17	⊢ 𝑀 = inf(𝑊, ℝ, < )

Assertion

Ref	Expression
ovolicc2lem4	⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Distinct variable groups:   𝑡,𝑛,𝑢,𝐴   𝐵,𝑛,𝑡,𝑢   𝑡,𝐻   𝐶,𝑛,𝑡   𝑛,𝐹,𝑡   𝑛,𝐾,𝑡,𝑢   𝑛,𝐺,𝑡   𝑛,𝑀,𝑡   𝑛,𝑊   𝜑,𝑛,𝑡   𝑇,𝑛,𝑡   𝑈,𝑛,𝑡,𝑢

Allowed substitution hints:   𝜑(𝑢)   𝐶(𝑢)   𝑆(𝑢,𝑡,𝑛)   𝑇(𝑢)   𝐹(𝑢)   𝐺(𝑢)   𝐻(𝑢,𝑛)   𝑀(𝑢)   𝑊(𝑢,𝑡)

Proof of Theorem ovolicc2lem4

Dummy variables 𝑚 𝑥 𝑦 𝑧 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		arch 11490	. . . . 5 ⊢ (𝑥 ∈ ℝ → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
2	1	ad2antlr 698	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥) → ∃𝑧 ∈ ℕ 𝑥 < 𝑧)
3		df-ima 5262	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺 ∘ 𝐾) “ (1...𝑀)) = ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀))
4		ovolicc2.8	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐺:𝑈⟶ℕ)
5		nnuz 11924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℕ = (ℤ≥‘1)
6		ovolicc2.15	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐾 = seq1((𝐻 ∘ 1st ), (ℕ × {𝐶}))
7		1zzd 11609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 1 ∈ ℤ)
8		ovolicc2.14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐶 ∈ 𝑇)
9		ovolicc2.11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐻:𝑇⟶𝑇)
10	5, 6, 7, 8, 9	algrf 15493	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾:ℕ⟶𝑇)
11		ovolicc2.10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑇 = {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅}
12		ssrab2 3834	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑢 ∈ 𝑈 ∣ (𝑢 ∩ (𝐴[,]𝐵)) ≠ ∅} ⊆ 𝑈
13	11, 12	eqsstri 3782	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑇 ⊆ 𝑈
14		fss 6196	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑇 ⊆ 𝑈) → 𝐾:ℕ⟶𝑈)
15	10, 13, 14	sylancl 566	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐾:ℕ⟶𝑈)
16		fco 6198	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:𝑈⟶ℕ ∧ 𝐾:ℕ⟶𝑈) → (𝐺 ∘ 𝐾):ℕ⟶ℕ)
17	4, 15, 16	syl2anc 565	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∘ 𝐾):ℕ⟶ℕ)
18		elfznn 12576	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
19	18	ssriv 3754	. . . . . . . . . . . . . . . . . 18 ⊢ (1...𝑀) ⊆ ℕ
20		fssres 6210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 ∘ 𝐾):ℕ⟶ℕ ∧ (1...𝑀) ⊆ ℕ) → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
21	17, 19, 20	sylancl 566	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ)
22		frn 6193	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ → ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)) ⊆ ℕ)
23	21, 22	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)) ⊆ ℕ)
24	3, 23	syl5eqss 3796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ ℕ)
25		nnssre 11225	. . . . . . . . . . . . . . 15 ⊢ ℕ ⊆ ℝ
26	24, 25	syl6ss 3762	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ ℝ)
27	26	ad3antrrr 701	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ ℝ)
28		simpr 471	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀)))
29	27, 28	sseldd 3751	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
30		simpllr 752	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑥 ∈ ℝ)
31		nnre 11228	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℝ)
32	31	ad2antlr 698	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
33		lelttr 10329	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 ≤ 𝑥 ∧ 𝑥 < 𝑧) → 𝑦 < 𝑧))
34	29, 30, 32, 33	syl3anc 1475	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → ((𝑦 ≤ 𝑥 ∧ 𝑥 < 𝑧) → 𝑦 < 𝑧))
35	34	ancomsd 456	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → ((𝑥 < 𝑧 ∧ 𝑦 ≤ 𝑥) → 𝑦 < 𝑧))
36	35	expdimp 440	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) ∧ 𝑥 < 𝑧) → (𝑦 ≤ 𝑥 → 𝑦 < 𝑧))
37	36	an32s 623	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (𝑦 ≤ 𝑥 → 𝑦 < 𝑧))
38	37	ralimdva 3110	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ 𝑥 < 𝑧) → (∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥 → ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧))
39	38	impancom 439	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑧 ∈ ℕ) ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧))
40	39	an32s 623	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥) ∧ 𝑧 ∈ ℕ) → (𝑥 < 𝑧 → ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧))
41	40	reximdva 3164	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥) → (∃𝑧 ∈ ℕ 𝑥 < 𝑧 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧))
42	2, 41	mpd 15	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥) → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)
43		fzfid 12979	. . . . 5 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
44		fvres 6348	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1...𝑀) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺 ∘ 𝐾)‘𝑖))
45	44	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = ((𝐺 ∘ 𝐾)‘𝑖))
46		fvco3 6417	. . . . . . . . . . . . . . 15 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑖 ∈ ℕ) → ((𝐺 ∘ 𝐾)‘𝑖) = (𝐺‘(𝐾‘𝑖)))
47	10, 18, 46	syl2an 575	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺 ∘ 𝐾)‘𝑖) = (𝐺‘(𝐾‘𝑖)))
48	45, 47	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾‘𝑖)))
49	48	adantrr 688	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (𝐺‘(𝐾‘𝑖)))
50		fvres 6348	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (1...𝑀) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺 ∘ 𝐾)‘𝑗))
51	50	ad2antll 700	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) = ((𝐺 ∘ 𝐾)‘𝑗))
52		elfznn 12576	. . . . . . . . . . . . . . 15 ⊢ (𝑗 ∈ (1...𝑀) → 𝑗 ∈ ℕ)
53	52	adantl 467	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀)) → 𝑗 ∈ ℕ)
54		fvco3 6417	. . . . . . . . . . . . . 14 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑗 ∈ ℕ) → ((𝐺 ∘ 𝐾)‘𝑗) = (𝐺‘(𝐾‘𝑗)))
55	10, 53, 54	syl2an 575	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺 ∘ 𝐾)‘𝑗) = (𝐺‘(𝐾‘𝑗)))
56	51, 55	eqtrd 2804	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) = (𝐺‘(𝐾‘𝑗)))
57	49, 56	eqeq12d 2785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) ↔ (𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑗))))
58		fveq2 6332	. . . . . . . . . . . . 13 ⊢ ((𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑗)) → (𝐹‘(𝐺‘(𝐾‘𝑖))) = (𝐹‘(𝐺‘(𝐾‘𝑗))))
59	58	fveq2d 6336	. . . . . . . . . . . 12 ⊢ ((𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑗)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑗)))))
60	19	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1...𝑀) ⊆ ℕ)
61		elfznn 12576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ∈ ℕ)
62	61	ad2antlr 698	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑛 ∈ ℕ)
63	62	nnred 11236	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑛 ∈ ℝ)
64		ovolicc2.16	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑊 = {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)}
65		ssrab2 3834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑛 ∈ ℕ ∣ 𝐵 ∈ (𝐾‘𝑛)} ⊆ ℕ
66	64, 65	eqsstri 3782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑊 ⊆ ℕ
67	66, 25	sstri 3759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑊 ⊆ ℝ
68		ovolicc2.17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑀 = inf(𝑊, ℝ, < )
69	66, 5	sseqtri 3784	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑊 ⊆ (ℤ≥‘1)
70		1z 11608	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 1 ∈ ℤ
71	5	uzinf 12971	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (1 ∈ ℤ → ¬ ℕ ∈ Fin)
72	70, 71	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ¬ ℕ ∈ Fin
73		ovolicc2.6	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → 𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin))
74		elin 3945	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑈 ∈ (𝒫 ran ((,) ∘ 𝐹) ∩ Fin) ↔ (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
75	73, 74	sylib 208	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝑈 ∈ 𝒫 ran ((,) ∘ 𝐹) ∧ 𝑈 ∈ Fin))
76	75	simprd 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑈 ∈ Fin)
77		ssfi 8335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑈 ∈ Fin ∧ 𝑇 ⊆ 𝑈) → 𝑇 ∈ Fin)
78	76, 13, 77	sylancl 566	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑇 ∈ Fin)
79	78	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝑇 ∈ Fin)
80	10	adantr 466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐾:ℕ⟶𝑇)
81		fveq2 6332	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝐾‘𝑖) = (𝐾‘𝑗) → (𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑗)))
82	81	fveq2d 6336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝐾‘𝑖) = (𝐾‘𝑗) → (𝐹‘(𝐺‘(𝐾‘𝑖))) = (𝐹‘(𝐺‘(𝐾‘𝑗))))
83	82	fveq2d 6336	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐾‘𝑖) = (𝐾‘𝑗) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑗)))))
84		simpll 742	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝜑)
85		simprl 746	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ ℕ)
86		ral0 4215	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ∀𝑚 ∈ ∅ 𝑛 ≤ 𝑚
87		simplr 744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑊 = ∅)
88	87	raleqdv 3292	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚 ↔ ∀𝑚 ∈ ∅ 𝑛 ≤ 𝑚))
89	86, 88	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚)
90	89	ralrimivw 3115	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ∀𝑛 ∈ ℕ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚)
91		rabid2 3266	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ↔ ∀𝑛 ∈ ℕ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚)
92	90, 91	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ℕ = {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
93	85, 92	eleqtrd 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
94		simprr 748	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ ℕ)
95	94, 92	eleqtrd 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
96		ovolicc.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝐴 ∈ ℝ)
97		ovolicc.2	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝐵 ∈ ℝ)
98		ovolicc.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
99		ovolicc2.4	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ 𝑆 = seq1( + , ((abs ∘ − ) ∘ 𝐹))
100		ovolicc2.5	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
101		ovolicc2.7	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ∪ 𝑈)
102		ovolicc2.9	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑈) → (((,) ∘ 𝐹)‘(𝐺‘𝑡)) = 𝑡)
103		ovolicc2.12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
104		ovolicc2.13	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
105	96, 97, 98, 99, 100, 73, 101, 4, 102, 11, 9, 103, 104, 8, 6, 64	ovolicc2lem3 23506	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝜑 ∧ (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑗))))))
106	84, 93, 95, 105	syl12anc 1473	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑗))))))
107	83, 106	syl5ibr 236	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝜑 ∧ 𝑊 = ∅) ∧ (𝑖 ∈ ℕ ∧ 𝑗 ∈ ℕ)) → ((𝐾‘𝑖) = (𝐾‘𝑗) → 𝑖 = 𝑗))
108	107	ralrimivva 3119	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾‘𝑖) = (𝐾‘𝑗) → 𝑖 = 𝑗))
109		dff13 6654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝐾:ℕ–1-1→𝑇 ↔ (𝐾:ℕ⟶𝑇 ∧ ∀𝑖 ∈ ℕ ∀𝑗 ∈ ℕ ((𝐾‘𝑖) = (𝐾‘𝑗) → 𝑖 = 𝑗)))
110	80, 108, 109	sylanbrc 564	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑊 = ∅) → 𝐾:ℕ–1-1→𝑇)
111		f1domg 8128	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑇 ∈ Fin → (𝐾:ℕ–1-1→𝑇 → ℕ ≼ 𝑇))
112	79, 110, 111	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ℕ ≼ 𝑇)
113		domfi 8336	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑇 ∈ Fin ∧ ℕ ≼ 𝑇) → ℕ ∈ Fin)
114	79, 112, 113	syl2anc 565	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑊 = ∅) → ℕ ∈ Fin)
115	114	ex 397	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝑊 = ∅ → ℕ ∈ Fin))
116	115	necon3bd 2956	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (¬ ℕ ∈ Fin → 𝑊 ≠ ∅))
117	72, 116	mpi 20	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑊 ≠ ∅)
118		infssuzcl 11974	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊 ⊆ (ℤ≥‘1) ∧ 𝑊 ≠ ∅) → inf(𝑊, ℝ, < ) ∈ 𝑊)
119	69, 117, 118	sylancr 567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → inf(𝑊, ℝ, < ) ∈ 𝑊)
120	68, 119	syl5eqel 2853	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑀 ∈ 𝑊)
121	67, 120	sseldi 3748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ ℝ)
122	121	ad2antrr 697	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑀 ∈ ℝ)
123	67	sseli 3746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ 𝑊 → 𝑚 ∈ ℝ)
124	123	adantl 467	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑚 ∈ ℝ)
125		elfzle2 12551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (1...𝑀) → 𝑛 ≤ 𝑀)
126	125	ad2antlr 698	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑛 ≤ 𝑀)
127		infssuzle 11973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑊 ⊆ (ℤ≥‘1) ∧ 𝑚 ∈ 𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑚)
128	69, 127	mpan 662	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ 𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑚)
129	68, 128	syl5eqbr 4819	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ 𝑊 → 𝑀 ≤ 𝑚)
130	129	adantl 467	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑀 ≤ 𝑚)
131	63, 122, 124, 126, 130	letrd 10395	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑀)) ∧ 𝑚 ∈ 𝑊) → 𝑛 ≤ 𝑚)
132	131	ralrimiva 3114	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑀)) → ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚)
133	60, 132	ssrabdv 3828	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
134	133	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (1...𝑀) ⊆ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
135		simprl 746	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ (1...𝑀))
136	134, 135	sseldd 3751	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
137		simprr 748	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ (1...𝑀))
138	134, 137	sseldd 3751	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚})
139	136, 138	jca 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚} ∧ 𝑗 ∈ {𝑛 ∈ ℕ ∣ ∀𝑚 ∈ 𝑊 𝑛 ≤ 𝑚}))
140	139, 105	syldan 571	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → (𝑖 = 𝑗 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑗))))))
141	59, 140	syl5ibr 236	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑗)) → 𝑖 = 𝑗))
142	57, 141	sylbid 230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑖 ∈ (1...𝑀) ∧ 𝑗 ∈ (1...𝑀))) → ((((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
143	142	ralrimivva 3119	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ (1...𝑀)∀𝑗 ∈ (1...𝑀)((((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗))
144		dff13 6654	. . . . . . . . 9 ⊢ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ ↔ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)⟶ℕ ∧ ∀𝑖 ∈ (1...𝑀)∀𝑗 ∈ (1...𝑀)((((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑖) = (((𝐺 ∘ 𝐾) ↾ (1...𝑀))‘𝑗) → 𝑖 = 𝑗)))
145	21, 143, 144	sylanbrc 564	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ)
146		f1f1orn 6289	. . . . . . . 8 ⊢ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1→ℕ → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)))
147	145, 146	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)))
148		f1oeq3 6270	. . . . . . . 8 ⊢ (((𝐺 ∘ 𝐾) “ (1...𝑀)) = ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)) → (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺 ∘ 𝐾) “ (1...𝑀)) ↔ ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀))))
149	3, 148	ax-mp 5	. . . . . . 7 ⊢ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺 ∘ 𝐾) “ (1...𝑀)) ↔ ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→ran ((𝐺 ∘ 𝐾) ↾ (1...𝑀)))
150	147, 149	sylibr 224	. . . . . 6 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺 ∘ 𝐾) “ (1...𝑀)))
151		f1ofo 6285	. . . . . 6 ⊢ (((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((𝐺 ∘ 𝐾) “ (1...𝑀)) → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺 ∘ 𝐾) “ (1...𝑀)))
152	150, 151	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺 ∘ 𝐾) “ (1...𝑀)))
153		fofi 8407	. . . . 5 ⊢ (((1...𝑀) ∈ Fin ∧ ((𝐺 ∘ 𝐾) ↾ (1...𝑀)):(1...𝑀)–onto→((𝐺 ∘ 𝐾) “ (1...𝑀))) → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ∈ Fin)
154	43, 152, 153	syl2anc 565	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ∈ Fin)
155		fimaxre2 11170	. . . 4 ⊢ ((((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ ℝ ∧ ((𝐺 ∘ 𝐾) “ (1...𝑀)) ∈ Fin) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥)
156	26, 154, 155	syl2anc 565	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ≤ 𝑥)
157	42, 156	r19.29a 3225	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℕ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)
158	97, 96	resubcld 10659	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ)
159	158	rexrd 10290	. . . 4 ⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ*)
160	159	adantr 466	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵 − 𝐴) ∈ ℝ*)
161		fzfid 12979	. . . . . 6 ⊢ (𝜑 → (1...𝑧) ∈ Fin)
162		elfznn 12576	. . . . . . . . 9 ⊢ (𝑗 ∈ (1...𝑧) → 𝑗 ∈ ℕ)
163		eqid 2770	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹)
164	163	ovolfsf 23458	. . . . . . . . . . 11 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
165	100, 164	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞))
166	165	ffvelrnda 6502	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
167	162, 166	sylan2 572	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞))
168		elrege0 12484	. . . . . . . 8 ⊢ ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ (0[,)+∞) ↔ ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
169	167, 168	sylib 208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑧)) → ((((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ ∧ 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗)))
170	169	simpld 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
171	161, 170	fsumrecl 14672	. . . . 5 ⊢ (𝜑 → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
172	171	adantr 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
173	172	rexrd 10290	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ*)
174	163, 99	ovolsf 23459	. . . . . . . . 9 ⊢ (𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞))
175	100, 174	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞))
176		frn 6193	. . . . . . . 8 ⊢ (𝑆:ℕ⟶(0[,)+∞) → ran 𝑆 ⊆ (0[,)+∞))
177	175, 176	syl 17	. . . . . . 7 ⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞))
178		rge0ssre 12486	. . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ
179	177, 178	syl6ss 3762	. . . . . 6 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ)
180		ressxr 10284	. . . . . 6 ⊢ ℝ ⊆ ℝ*
181	179, 180	syl6ss 3762	. . . . 5 ⊢ (𝜑 → ran 𝑆 ⊆ ℝ*)
182		supxrcl 12349	. . . . 5 ⊢ (ran 𝑆 ⊆ ℝ* → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
183	181, 182	syl 17	. . . 4 ⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
184	183	adantr 466	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → sup(ran 𝑆, ℝ*, < ) ∈ ℝ*)
185	158	adantr 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵 − 𝐴) ∈ ℝ)
186	24	sselda 3750	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑗 ∈ ℕ)
187	178, 166	sseldi 3748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
188	186, 187	syldan 571	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
189	154, 188	fsumrecl 14672	. . . . 5 ⊢ (𝜑 → Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
190	189	adantr 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
191		inss2 3980	. . . . . . . . . . 11 ⊢ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)
192		fss 6196	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ)) ⊆ (ℝ × ℝ)) → 𝐹:ℕ⟶(ℝ × ℝ))
193	100, 191, 192	sylancl 566	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶(ℝ × ℝ))
194	66, 120	sseldi 3748	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℕ)
195	15, 194	ffvelrnd 6503	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾‘𝑀) ∈ 𝑈)
196	4, 195	ffvelrnd 6503	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘(𝐾‘𝑀)) ∈ ℕ)
197	193, 196	ffvelrnd 6503	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐾‘𝑀))) ∈ (ℝ × ℝ))
198		xp2nd 7347	. . . . . . . . 9 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑀))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) ∈ ℝ)
199	197, 198	syl 17	. . . . . . . 8 ⊢ (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) ∈ ℝ)
200	13, 8	sseldi 3748	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
201	4, 200	ffvelrnd 6503	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝐶) ∈ ℕ)
202	193, 201	ffvelrnd 6503	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘(𝐺‘𝐶)) ∈ (ℝ × ℝ))
203		xp1st 7346	. . . . . . . . 9 ⊢ ((𝐹‘(𝐺‘𝐶)) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘𝐶))) ∈ ℝ)
204	202, 203	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1st ‘(𝐹‘(𝐺‘𝐶))) ∈ ℝ)
205	199, 204	resubcld 10659	. . . . . . 7 ⊢ (𝜑 → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) ∈ ℝ)
206		fveq2 6332	. . . . . . . . . 10 ⊢ (𝑗 = (𝐺‘(𝐾‘𝑖)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾‘𝑖))))
207	187	recnd 10269	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
208	186, 207	syldan 571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
209	206, 43, 150, 48, 208	fsumf1o 14661	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) = Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾‘𝑖))))
210	100	adantr 466	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
211	4	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐺:𝑈⟶ℕ)
212		ffvelrn 6500	. . . . . . . . . . . . 13 ⊢ ((𝐾:ℕ⟶𝑈 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) ∈ 𝑈)
213	15, 18, 212	syl2an 575	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝑈)
214	211, 213	ffvelrnd 6503	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘(𝐾‘𝑖)) ∈ ℕ)
215	163	ovolfsval 23457	. . . . . . . . . . 11 ⊢ ((𝐹:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ (𝐺‘(𝐾‘𝑖)) ∈ ℕ) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾‘𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
216	210, 214, 215	syl2anc 565	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾‘𝑖))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
217	216	sumeq2dv 14640	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)(((abs ∘ − ) ∘ 𝐹)‘(𝐺‘(𝐾‘𝑖))) = Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
218	193	adantr 466	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝐹:ℕ⟶(ℝ × ℝ))
219	4	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝐺:𝑈⟶ℕ)
220	15	ffvelrnda 6502	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) ∈ 𝑈)
221	219, 220	ffvelrnd 6503	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐺‘(𝐾‘𝑖)) ∈ ℕ)
222	218, 221	ffvelrnd 6503	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘𝑖))) ∈ (ℝ × ℝ))
223		xp2nd 7347	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑖))) ∈ (ℝ × ℝ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
224	222, 223	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
225	18, 224	sylan2 572	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
226	225	recnd 10269	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℂ)
227	193	adantr 466	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝐹:ℕ⟶(ℝ × ℝ))
228	227, 214	ffvelrnd 6503	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐹‘(𝐺‘(𝐾‘𝑖))) ∈ (ℝ × ℝ))
229		xp1st 7346	. . . . . . . . . . . . 13 ⊢ ((𝐹‘(𝐺‘(𝐾‘𝑖))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
230	228, 229	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
231	230	recnd 10269	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℂ)
232	43, 226, 231	fsumsub 14726	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) = (Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
233	69, 120	sseldi 3748	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
234		fveq2 6332	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑀 → (𝐾‘𝑖) = (𝐾‘𝑀))
235	234	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑀 → (𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘𝑀)))
236	235	fveq2d 6336	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑀 → (𝐹‘(𝐺‘(𝐾‘𝑖))) = (𝐹‘(𝐺‘(𝐾‘𝑀))))
237	236	fveq2d 6336	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑀 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))))
238	233, 226, 237	fsumm1 14687	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) + (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀))))))
239		fzfid 12979	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1...(𝑀 − 1)) ∈ Fin)
240		elfznn 12576	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...(𝑀 − 1)) → 𝑖 ∈ ℕ)
241	240, 224	sylan2 572	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
242	239, 241	fsumrecl 14672	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ)
243	242	recnd 10269	. . . . . . . . . . . . 13 ⊢ (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℂ)
244	199	recnd 10269	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) ∈ ℂ)
245	243, 244	addcomd 10439	. . . . . . . . . . . 12 ⊢ (𝜑 → (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) + (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀))))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
246	238, 245	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
247		fveq2 6332	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝐾‘𝑖) = (𝐾‘1))
248	247	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 1 → (𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘1)))
249	248	fveq2d 6336	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (𝐹‘(𝐺‘(𝐾‘𝑖))) = (𝐹‘(𝐺‘(𝐾‘1))))
250	249	fveq2d 6336	. . . . . . . . . . . . 13 ⊢ (𝑖 = 1 → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))))
251	233, 231, 250	fsum1p 14689	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = ((1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) + Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
252	5, 6, 7, 8	algr0 15492	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐾‘1) = 𝐶)
253	252	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐺‘(𝐾‘1)) = (𝐺‘𝐶))
254	253	fveq2d 6336	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(𝐺‘(𝐾‘1))) = (𝐹‘(𝐺‘𝐶)))
255	254	fveq2d 6336	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) = (1st ‘(𝐹‘(𝐺‘𝐶))))
256	7	peano2zd 11686	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 + 1) ∈ ℤ)
257	194	nnzd 11682	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑀 ∈ ℤ)
258		fzp1ss 12598	. . . . . . . . . . . . . . . . . 18 ⊢ (1 ∈ ℤ → ((1 + 1)...𝑀) ⊆ (1...𝑀))
259	70, 258	mp1i 13	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1 + 1)...𝑀) ⊆ (1...𝑀))
260	259	sselda 3750	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ((1 + 1)...𝑀)) → 𝑖 ∈ (1...𝑀))
261	260, 231	syldan 571	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ((1 + 1)...𝑀)) → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℂ)
262		fveq2 6332	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = (𝑗 + 1) → (𝐾‘𝑖) = (𝐾‘(𝑗 + 1)))
263	262	fveq2d 6336	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = (𝑗 + 1) → (𝐺‘(𝐾‘𝑖)) = (𝐺‘(𝐾‘(𝑗 + 1))))
264	263	fveq2d 6336	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 + 1) → (𝐹‘(𝐺‘(𝐾‘𝑖))) = (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
265	264	fveq2d 6336	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 + 1) → (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
266	7, 256, 257, 261, 265	fsumshftm 14719	. . . . . . . . . . . . . 14 ⊢ (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))))
267		ax-1cn 10195	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℂ
268	267, 267	pncan3oi 10498	. . . . . . . . . . . . . . . . 17 ⊢ ((1 + 1) − 1) = 1
269	268	oveq1i 6802	. . . . . . . . . . . . . . . 16 ⊢ (((1 + 1) − 1)...(𝑀 − 1)) = (1...(𝑀 − 1))
270	269	sumeq1i 14635	. . . . . . . . . . . . . . 15 ⊢ Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))))
271		fvoveq1 6815	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = 𝑖 → (𝐾‘(𝑗 + 1)) = (𝐾‘(𝑖 + 1)))
272	271	fveq2d 6336	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑖 → (𝐺‘(𝐾‘(𝑗 + 1))) = (𝐺‘(𝐾‘(𝑖 + 1))))
273	272	fveq2d 6336	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑖 → (𝐹‘(𝐺‘(𝐾‘(𝑗 + 1)))) = (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
274	273	fveq2d 6336	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑖 → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
275	274	cbvsumv 14633	. . . . . . . . . . . . . . 15 ⊢ Σ𝑗 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
276	270, 275	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ Σ𝑗 ∈ (((1 + 1) − 1)...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑗 + 1))))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))
277	266, 276	syl6eq 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))
278	255, 277	oveq12d 6810	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1st ‘(𝐹‘(𝐺‘(𝐾‘1)))) + Σ𝑖 ∈ ((1 + 1)...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) = ((1st ‘(𝐹‘(𝐺‘𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
279	251, 278	eqtrd 2804	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) = ((1st ‘(𝐹‘(𝐺‘𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
280	246, 279	oveq12d 6810	. . . . . . . . . 10 ⊢ (𝜑 → (Σ𝑖 ∈ (1...𝑀)(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...𝑀)(1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) − ((1st ‘(𝐹‘(𝐺‘𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
281	204	recnd 10269	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘(𝐹‘(𝐺‘𝐶))) ∈ ℂ)
282		peano2nn 11233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ ℕ → (𝑖 + 1) ∈ ℕ)
283		ffvelrn 6500	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾:ℕ⟶𝑈 ∧ (𝑖 + 1) ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
284	15, 282, 283	syl2an 575	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
285	219, 284	ffvelrnd 6503	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐺‘(𝐾‘(𝑖 + 1))) ∈ ℕ)
286	218, 285	ffvelrnd 6503	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ))
287		xp1st 7346	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))) ∈ (ℝ × ℝ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
288	286, 287	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
289	240, 288	sylan2 572	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
290	239, 289	fsumrecl 14672	. . . . . . . . . . . 12 ⊢ (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℝ)
291	290	recnd 10269	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ∈ ℂ)
292	244, 243, 281, 291	addsub4d 10640	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) + Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) − ((1st ‘(𝐹‘(𝐺‘𝐶))) + Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
293	232, 280, 292	3eqtrd 2808	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))) = (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
294	209, 217, 293	3eqtrd 2808	. . . . . . . 8 ⊢ (𝜑 → Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
295	294, 189	eqeltrrd 2850	. . . . . . 7 ⊢ (𝜑 → (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))) ∈ ℝ)
296		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑀 → (𝐾‘𝑛) = (𝐾‘𝑀))
297	296	eleq2d 2835	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑀 → (𝐵 ∈ (𝐾‘𝑛) ↔ 𝐵 ∈ (𝐾‘𝑀)))
298	297, 64	elrab2 3516	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝑊 ↔ (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾‘𝑀)))
299	120, 298	sylib 208	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝐵 ∈ (𝐾‘𝑀)))
300	299	simprd 477	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ (𝐾‘𝑀))
301	96, 97, 98, 99, 100, 73, 101, 4, 102	ovolicc2lem1 23504	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝐾‘𝑀) ∈ 𝑈) → (𝐵 ∈ (𝐾‘𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) < 𝐵 ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))))))
302	195, 301	mpdan 659	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ∈ (𝐾‘𝑀) ↔ (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) < 𝐵 ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))))))
303	300, 302	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) < 𝐵 ∧ 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀))))))
304	303	simp3d 1137	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))))
305	96, 97, 98, 99, 100, 73, 101, 4, 102	ovolicc2lem1 23504	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝑈) → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝐶))) < 𝐴 ∧ 𝐴 < (2nd ‘(𝐹‘(𝐺‘𝐶))))))
306	200, 305	mpdan 659	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝐶))) < 𝐴 ∧ 𝐴 < (2nd ‘(𝐹‘(𝐺‘𝐶))))))
307	104, 306	mpbid 222	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘𝐶))) < 𝐴 ∧ 𝐴 < (2nd ‘(𝐹‘(𝐺‘𝐶)))))
308	307	simp2d 1136	. . . . . . . . 9 ⊢ (𝜑 → (1st ‘(𝐹‘(𝐺‘𝐶))) < 𝐴)
309	97, 204, 199, 96, 304, 308	lt2subd 10852	. . . . . . . 8 ⊢ (𝜑 → (𝐵 − 𝐴) < ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))))
310	158, 205, 309	ltled 10386	. . . . . . 7 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))))
311	240	adantl 467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℕ)
312		simpr 471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ (1...(𝑀 − 1)))
313	257	adantr 466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℤ)
314		elfzm11 12617	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
315	70, 313, 314	sylancr 567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ (1...(𝑀 − 1)) ↔ (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖 ∧ 𝑖 < 𝑀)))
316	312, 315	mpbid 222	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℤ ∧ 1 ≤ 𝑖 ∧ 𝑖 < 𝑀))
317	316	simp3d 1137	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 < 𝑀)
318	311	nnred 11236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑖 ∈ ℝ)
319	121	adantr 466	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → 𝑀 ∈ ℝ)
320	318, 319	ltnled 10385	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑖))
321	317, 320	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑀 ≤ 𝑖)
322		infssuzle 11973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑊 ⊆ (ℤ≥‘1) ∧ 𝑖 ∈ 𝑊) → inf(𝑊, ℝ, < ) ≤ 𝑖)
323	69, 322	mpan 662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ 𝑊 → inf(𝑊, ℝ, < ) ≤ 𝑖)
324	68, 323	syl5eqbr 4819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑊 → 𝑀 ≤ 𝑖)
325	321, 324	nsyl 137	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ¬ 𝑖 ∈ 𝑊)
326	311, 325	jca 495	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝑖 ∈ ℕ ∧ ¬ 𝑖 ∈ 𝑊))
327	96, 97, 98, 99, 100, 73, 101, 4, 102, 11, 9, 103, 104, 8, 6, 64	ovolicc2lem2 23505	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖 ∈ ℕ ∧ ¬ 𝑖 ∈ 𝑊)) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵)
328	326, 327	syldan 571	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵)
329	328	iftrued 4231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))), 𝐵) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))))
330		fveq2 6332	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐾‘𝑖) → (𝐺‘𝑡) = (𝐺‘(𝐾‘𝑖)))
331	330	fveq2d 6336	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐾‘𝑖) → (𝐹‘(𝐺‘𝑡)) = (𝐹‘(𝐺‘(𝐾‘𝑖))))
332	331	fveq2d 6336	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐾‘𝑖) → (2nd ‘(𝐹‘(𝐺‘𝑡))) = (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))))
333	332	breq1d 4794	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐾‘𝑖) → ((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵 ↔ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵))
334	333, 332	ifbieq1d 4246	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐾‘𝑖) → if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) = if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))), 𝐵))
335		fveq2 6332	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐾‘𝑖) → (𝐻‘𝑡) = (𝐻‘(𝐾‘𝑖)))
336	334, 335	eleq12d 2843	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = (𝐾‘𝑖) → (if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡) ↔ if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))), 𝐵) ∈ (𝐻‘(𝐾‘𝑖))))
337	103	ralrimiva 3114	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
338	337	adantr 466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ∀𝑡 ∈ 𝑇 if((2nd ‘(𝐹‘(𝐺‘𝑡))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘𝑡))), 𝐵) ∈ (𝐻‘𝑡))
339		ffvelrn 6500	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐾:ℕ⟶𝑇 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) ∈ 𝑇)
340	10, 240, 339	syl2an 575	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘𝑖) ∈ 𝑇)
341	336, 338, 340	rspcdva 3464	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → if((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ≤ 𝐵, (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))), 𝐵) ∈ (𝐻‘(𝐾‘𝑖)))
342	329, 341	eqeltrrd 2850	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ (𝐻‘(𝐾‘𝑖)))
343	5, 6, 7, 8, 9	algrp1 15494	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾‘𝑖)))
344	240, 343	sylan2 572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) = (𝐻‘(𝐾‘𝑖)))
345	342, 344	eleqtrrd 2852	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ (𝐾‘(𝑖 + 1)))
346	240, 284	sylan2 572	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (𝐾‘(𝑖 + 1)) ∈ 𝑈)
347	96, 97, 98, 99, 100, 73, 101, 4, 102	ovolicc2lem1 23504	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝐾‘(𝑖 + 1)) ∈ 𝑈) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ (𝐾‘(𝑖 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
348	346, 347	syldan 571	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ (𝐾‘(𝑖 + 1)) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
349	345, 348	mpbid 222	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∈ ℝ ∧ (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) ∧ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
350	349	simp2d 1136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) < (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))))
351	289, 241, 350	ltled 10386	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...(𝑀 − 1))) → (1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ (2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))))
352	239, 289, 241, 351	fsumle 14737	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))))
353	242, 290	subge0d 10818	. . . . . . . . 9 ⊢ (𝜑 → (0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ↔ Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))) ≤ Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖))))))
354	352, 353	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → 0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))
355	242, 290	resubcld 10659	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ∈ ℝ)
356	205, 355	addge01d 10816	. . . . . . . 8 ⊢ (𝜑 → (0 ≤ (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))) ↔ ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1)))))))))
357	354, 356	mpbid 222	. . . . . . 7 ⊢ (𝜑 → ((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
358	158, 205, 295, 310, 357	letrd 10395	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ (((2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑀)))) − (1st ‘(𝐹‘(𝐺‘𝐶)))) + (Σ𝑖 ∈ (1...(𝑀 − 1))(2nd ‘(𝐹‘(𝐺‘(𝐾‘𝑖)))) − Σ𝑖 ∈ (1...(𝑀 − 1))(1st ‘(𝐹‘(𝐺‘(𝐾‘(𝑖 + 1))))))))
359	358, 294	breqtrrd 4812	. . . . 5 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
360	359	adantr 466	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵 − 𝐴) ≤ Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗))
361		fzfid 12979	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (1...𝑧) ∈ Fin)
362	170	adantlr 686	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℝ)
363	169	simprd 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
364	363	adantlr 686	. . . . 5 ⊢ (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → 0 ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑗))
365	24	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ ℕ)
366	365	sselda 3750	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑦 ∈ ℕ)
367	366	nnred 11236	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑦 ∈ ℝ)
368	31	ad2antlr 698	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑧 ∈ ℝ)
369		ltle 10327	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑦 < 𝑧 → 𝑦 ≤ 𝑧))
370	367, 368, 369	syl2anc 565	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (𝑦 < 𝑧 → 𝑦 ≤ 𝑧))
371	366, 5	syl6eleq 2859	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑦 ∈ (ℤ≥‘1))
372		nnz 11600	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
373	372	ad2antlr 698	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → 𝑧 ∈ ℤ)
374		elfz5 12540	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (ℤ≥‘1) ∧ 𝑧 ∈ ℤ) → (𝑦 ∈ (1...𝑧) ↔ 𝑦 ≤ 𝑧))
375	371, 373, 374	syl2anc 565	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (𝑦 ∈ (1...𝑧) ↔ 𝑦 ≤ 𝑧))
376	370, 375	sylibrd 249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ) ∧ 𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))) → (𝑦 < 𝑧 → 𝑦 ∈ (1...𝑧)))
377	376	ralimdva 3110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ) → (∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧 → ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧)))
378	377	impr 442	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
379		dfss3 3739	. . . . . 6 ⊢ (((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ (1...𝑧) ↔ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 ∈ (1...𝑧))
380	378, 379	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ((𝐺 ∘ 𝐾) “ (1...𝑀)) ⊆ (1...𝑧))
381	361, 362, 364, 380	fsumless 14734	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
382	185, 190, 172, 360, 381	letrd 10395	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵 − 𝐴) ≤ Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗))
383		eqidd 2771	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) = (((abs ∘ − ) ∘ 𝐹)‘𝑗))
384		simprl 746	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ ℕ)
385	384, 5	syl6eleq 2859	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑧 ∈ (ℤ≥‘1))
386	362	recnd 10269	. . . . . 6 ⊢ (((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) ∧ 𝑗 ∈ (1...𝑧)) → (((abs ∘ − ) ∘ 𝐹)‘𝑗) ∈ ℂ)
387	383, 385, 386	fsumser 14668	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧))
388	99	fveq1i 6333	. . . . 5 ⊢ (𝑆‘𝑧) = (seq1( + , ((abs ∘ − ) ∘ 𝐹))‘𝑧)
389	387, 388	syl6eqr 2822	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) = (𝑆‘𝑧))
390	181	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → ran 𝑆 ⊆ ℝ*)
391		ffn 6185	. . . . . . . 8 ⊢ (𝑆:ℕ⟶(0[,)+∞) → 𝑆 Fn ℕ)
392	175, 391	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 Fn ℕ)
393	392	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → 𝑆 Fn ℕ)
394		fnfvelrn 6499	. . . . . 6 ⊢ ((𝑆 Fn ℕ ∧ 𝑧 ∈ ℕ) → (𝑆‘𝑧) ∈ ran 𝑆)
395	393, 384, 394	syl2anc 565	. . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆‘𝑧) ∈ ran 𝑆)
396		supxrub 12358	. . . . 5 ⊢ ((ran 𝑆 ⊆ ℝ* ∧ (𝑆‘𝑧) ∈ ran 𝑆) → (𝑆‘𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
397	390, 395, 396	syl2anc 565	. . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝑆‘𝑧) ≤ sup(ran 𝑆, ℝ*, < ))
398	389, 397	eqbrtrd 4806	. . 3 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → Σ𝑗 ∈ (1...𝑧)(((abs ∘ − ) ∘ 𝐹)‘𝑗) ≤ sup(ran 𝑆, ℝ*, < ))
399	160, 173, 184, 382, 398	xrletrd 12197	. 2 ⊢ ((𝜑 ∧ (𝑧 ∈ ℕ ∧ ∀𝑦 ∈ ((𝐺 ∘ 𝐾) “ (1...𝑀))𝑦 < 𝑧)) → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))
400	157, 399	rexlimddv 3182	1 ⊢ (𝜑 → (𝐵 − 𝐴) ≤ sup(ran 𝑆, ℝ*, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144   ≠ wne 2942  ∀wral 3060  ∃wrex 3061  {crab 3064   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  ifcif 4223  𝒫 cpw 4295  {csn 4314  ∪ cuni 4572   class class class wbr 4784   × cxp 5247  ran crn 5250   ↾ cres 5251   “ cima 5252   ∘ ccom 5253   Fn wfn 6026  ⟶wf 6027  –1-1→wf1 6028  –onto→wfo 6029  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792  1^st c1st 7312  2^nd c2nd 7313   ≼ cdom 8106  Fincfn 8108  supcsup 8501  infcinf 8502  ℂcc 10135  ℝcr 10136  0cc0 10137  1c1 10138   + caddc 10140  +∞cpnf 10272  ℝ^*cxr 10274   < clt 10275   ≤ cle 10276   − cmin 10467  ℕcn 11221  ℤcz 11578  ℤ_≥cuz 11887  (,)cioo 12379  [,)cico 12381  [,]cicc 12382  ...cfz 12532  seqcseq 13007  abscabs 14181  Σcsu 14623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-pre-sup 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-sup 8503  df-inf 8504  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-div 10886  df-nn 11222  df-2 11280  df-3 11281  df-n0 11494  df-z 11579  df-uz 11888  df-rp 12035  df-ioo 12383  df-ico 12385  df-icc 12386  df-fz 12533  df-fzo 12673  df-seq 13008  df-exp 13067  df-hash 13321  df-cj 14046  df-re 14047  df-im 14048  df-sqrt 14182  df-abs 14183  df-clim 14426  df-sum 14624

This theorem is referenced by:  ovolicc2lem5  23508