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Theorem gsumbagdiaglem 19423
Description: Lemma for gsumbagdiag 19424. (Contributed by Mario Carneiro, 5-Jan-2015.)
Hypotheses
Ref Expression
psrbag.d 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
psrbagconf1o.1 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
gsumbagdiag.i (𝜑𝐼𝑉)
gsumbagdiag.f (𝜑𝐹𝐷)
Assertion
Ref Expression
gsumbagdiaglem ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
Distinct variable groups:   𝑥,𝑓,𝑦,𝐹   𝑥,𝑉,𝑦   𝑓,𝐼,𝑥,𝑦   𝑥,𝑆   𝑥,𝐷,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐷(𝑓)   𝑆(𝑦,𝑓)   𝑉(𝑓)

Proof of Theorem gsumbagdiaglem
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 811 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})
2 breq1 4688 . . . . . 6 (𝑥 = 𝑌 → (𝑥𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑌𝑟 ≤ (𝐹𝑓𝑋)))
32elrab 3396 . . . . 5 (𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)} ↔ (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
41, 3sylib 208 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝐷𝑌𝑟 ≤ (𝐹𝑓𝑋)))
54simpld 474 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝐷)
64simprd 478 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟 ≤ (𝐹𝑓𝑋))
7 gsumbagdiag.i . . . . . . 7 (𝜑𝐼𝑉)
87adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐼𝑉)
9 gsumbagdiag.f . . . . . . 7 (𝜑𝐹𝐷)
109adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹𝐷)
11 simprl 809 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑆)
12 breq1 4688 . . . . . . . . . 10 (𝑦 = 𝑋 → (𝑦𝑟𝐹𝑋𝑟𝐹))
13 psrbagconf1o.1 . . . . . . . . . 10 𝑆 = {𝑦𝐷𝑦𝑟𝐹}
1412, 13elrab2 3399 . . . . . . . . 9 (𝑋𝑆 ↔ (𝑋𝐷𝑋𝑟𝐹))
1511, 14sylib 208 . . . . . . . 8 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝐷𝑋𝑟𝐹))
1615simpld 474 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝐷)
17 psrbag.d . . . . . . . 8 𝐷 = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
1817psrbagf 19413 . . . . . . 7 ((𝐼𝑉𝑋𝐷) → 𝑋:𝐼⟶ℕ0)
198, 16, 18syl2anc 694 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋:𝐼⟶ℕ0)
2015simprd 478 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟𝐹)
2117psrbagcon 19419 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝐷𝑋:𝐼⟶ℕ0𝑋𝑟𝐹)) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
228, 10, 19, 20, 21syl13anc 1368 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝐹𝑓𝑋) ∈ 𝐷 ∧ (𝐹𝑓𝑋) ∘𝑟𝐹))
2322simprd 478 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∘𝑟𝐹)
2417psrbagf 19413 . . . . . 6 ((𝐼𝑉𝑌𝐷) → 𝑌:𝐼⟶ℕ0)
258, 5, 24syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌:𝐼⟶ℕ0)
2622simpld 474 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) ∈ 𝐷)
2717psrbagf 19413 . . . . . 6 ((𝐼𝑉 ∧ (𝐹𝑓𝑋) ∈ 𝐷) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
288, 26, 27syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋):𝐼⟶ℕ0)
2917psrbagf 19413 . . . . . 6 ((𝐼𝑉𝐹𝐷) → 𝐹:𝐼⟶ℕ0)
308, 10, 29syl2anc 694 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹:𝐼⟶ℕ0)
31 nn0re 11339 . . . . . . 7 (𝑢 ∈ ℕ0𝑢 ∈ ℝ)
32 nn0re 11339 . . . . . . 7 (𝑣 ∈ ℕ0𝑣 ∈ ℝ)
33 nn0re 11339 . . . . . . 7 (𝑤 ∈ ℕ0𝑤 ∈ ℝ)
34 letr 10169 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ ∧ 𝑤 ∈ ℝ) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3531, 32, 33, 34syl3an 1408 . . . . . 6 ((𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
3635adantl 481 . . . . 5 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ (𝑢 ∈ ℕ0𝑣 ∈ ℕ0𝑤 ∈ ℕ0)) → ((𝑢𝑣𝑣𝑤) → 𝑢𝑤))
378, 25, 28, 30, 36caoftrn 6974 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → ((𝑌𝑟 ≤ (𝐹𝑓𝑋) ∧ (𝐹𝑓𝑋) ∘𝑟𝐹) → 𝑌𝑟𝐹))
386, 23, 37mp2and 715 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑟𝐹)
39 breq1 4688 . . . 4 (𝑦 = 𝑌 → (𝑦𝑟𝐹𝑌𝑟𝐹))
4039, 13elrab2 3399 . . 3 (𝑌𝑆 ↔ (𝑌𝐷𝑌𝑟𝐹))
415, 38, 40sylanbrc 699 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌𝑆)
4219ffvelrnda 6399 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) ∈ ℕ0)
4325ffvelrnda 6399 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) ∈ ℕ0)
4430ffvelrnda 6399 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) ∈ ℕ0)
45 nn0re 11339 . . . . . . . 8 ((𝑋𝑧) ∈ ℕ0 → (𝑋𝑧) ∈ ℝ)
46 nn0re 11339 . . . . . . . 8 ((𝑌𝑧) ∈ ℕ0 → (𝑌𝑧) ∈ ℝ)
47 nn0re 11339 . . . . . . . 8 ((𝐹𝑧) ∈ ℕ0 → (𝐹𝑧) ∈ ℝ)
48 leaddsub2 10543 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
49 leaddsub 10542 . . . . . . . . 9 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → (((𝑋𝑧) + (𝑌𝑧)) ≤ (𝐹𝑧) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5048, 49bitr3d 270 . . . . . . . 8 (((𝑋𝑧) ∈ ℝ ∧ (𝑌𝑧) ∈ ℝ ∧ (𝐹𝑧) ∈ ℝ) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5145, 46, 47, 50syl3an 1408 . . . . . . 7 (((𝑋𝑧) ∈ ℕ0 ∧ (𝑌𝑧) ∈ ℕ0 ∧ (𝐹𝑧) ∈ ℕ0) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5242, 43, 44, 51syl3anc 1366 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
5352ralbidva 3014 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧)) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
54 ovexd 6720 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑋𝑧)) ∈ V)
5525feqmptd 6288 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 = (𝑧𝐼 ↦ (𝑌𝑧)))
56 ffn 6083 . . . . . . . 8 (𝐹:𝐼⟶ℕ0𝐹 Fn 𝐼)
5730, 56syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝐹 Fn 𝐼)
58 ffn 6083 . . . . . . . 8 (𝑋:𝐼⟶ℕ0𝑋 Fn 𝐼)
5919, 58syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 Fn 𝐼)
60 inidm 3855 . . . . . . 7 (𝐼𝐼) = 𝐼
61 eqidd 2652 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝐹𝑧) = (𝐹𝑧))
62 eqidd 2652 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑋𝑧) = (𝑋𝑧))
6357, 59, 8, 8, 60, 61, 62offval 6946 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑋) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑋𝑧))))
648, 43, 54, 55, 63ofrfval2 6957 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ ∀𝑧𝐼 (𝑌𝑧) ≤ ((𝐹𝑧) − (𝑋𝑧))))
65 ovexd 6720 . . . . . 6 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → ((𝐹𝑧) − (𝑌𝑧)) ∈ V)
6619feqmptd 6288 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 = (𝑧𝐼 ↦ (𝑋𝑧)))
67 ffn 6083 . . . . . . . 8 (𝑌:𝐼⟶ℕ0𝑌 Fn 𝐼)
6825, 67syl 17 . . . . . . 7 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑌 Fn 𝐼)
69 eqidd 2652 . . . . . . 7 (((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) ∧ 𝑧𝐼) → (𝑌𝑧) = (𝑌𝑧))
7057, 68, 8, 8, 60, 61, 69offval 6946 . . . . . 6 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝐹𝑓𝑌) = (𝑧𝐼 ↦ ((𝐹𝑧) − (𝑌𝑧))))
718, 42, 65, 66, 70ofrfval2 6957 . . . . 5 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑋𝑟 ≤ (𝐹𝑓𝑌) ↔ ∀𝑧𝐼 (𝑋𝑧) ≤ ((𝐹𝑧) − (𝑌𝑧))))
7253, 64, 713bitr4d 300 . . . 4 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑟 ≤ (𝐹𝑓𝑋) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
736, 72mpbid 222 . . 3 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋𝑟 ≤ (𝐹𝑓𝑌))
74 breq1 4688 . . . 4 (𝑥 = 𝑋 → (𝑥𝑟 ≤ (𝐹𝑓𝑌) ↔ 𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7574elrab 3396 . . 3 (𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)} ↔ (𝑋𝐷𝑋𝑟 ≤ (𝐹𝑓𝑌)))
7616, 73, 75sylanbrc 699 . 2 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → 𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)})
7741, 76jca 553 1 ((𝜑 ∧ (𝑋𝑆𝑌 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑋)})) → (𝑌𝑆𝑋 ∈ {𝑥𝐷𝑥𝑟 ≤ (𝐹𝑓𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231   class class class wbr 4685  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937  𝑟 cofr 6938  𝑚 cmap 7899  Fincfn 7997  cr 9973   + caddc 9977  cle 10113  cmin 10304  cn 11058  0cn0 11330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331
This theorem is referenced by:  gsumbagdiag  19424  psrass1lem  19425
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