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Theorem reprdifc 31014
Description: Express the representations as a sum of integers in a difference of sets using conditions on each of the indices. (Contributed by Thierry Arnoux, 27-Dec-2021.)
Hypotheses
Ref Expression
reprdifc.c 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}
reprdifc.a (𝜑𝐴 ⊆ ℕ)
reprdifc.b (𝜑𝐵 ⊆ ℕ)
reprdifc.m (𝜑𝑀 ∈ ℕ0)
reprdifc.s (𝜑𝑆 ∈ ℕ0)
Assertion
Ref Expression
reprdifc (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = 𝑥 ∈ (0..^𝑆)𝐶)
Distinct variable groups:   𝐴,𝑐,𝑥   𝐵,𝑐,𝑥   𝑀,𝑐,𝑥   𝑆,𝑐,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑐)   𝐶(𝑥,𝑐)

Proof of Theorem reprdifc
Dummy variables 𝑑 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . 3 𝑑𝜑
2 nfrab1 3261 . . 3 𝑑{𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}
3 nfcv 2902 . . 3 𝑑 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}
4 reprdifc.a . . . . . . . . . . 11 (𝜑𝐴 ⊆ ℕ)
5 reprdifc.m . . . . . . . . . . . 12 (𝜑𝑀 ∈ ℕ0)
65nn0zd 11672 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7 reprdifc.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ0)
84, 6, 7reprval 30997 . . . . . . . . . 10 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
98eleq2d 2825 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
10 rabid 3254 . . . . . . . . 9 (𝑑 ∈ {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
119, 10syl6bb 276 . . . . . . . 8 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀)))
1211anbi1d 743 . . . . . . 7 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ↔ ((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)))))
13 eldif 3725 . . . . . . . . . 10 (𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))))
1413anbi1i 733 . . . . . . . . 9 ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
15 an32 874 . . . . . . . . 9 (((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))))
1614, 15bitri 264 . . . . . . . 8 ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))))
1716a1i 11 . . . . . . 7 (𝜑 → ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ ((𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)))))
1812, 17bitr4d 271 . . . . . 6 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀)))
19 nnex 11218 . . . . . . . . . . . . . 14 ℕ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
21 reprdifc.b . . . . . . . . . . . . 13 (𝜑𝐵 ⊆ ℕ)
2220, 21ssexd 4957 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
23 ovexd 6843 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ∈ V)
24 elmapg 8036 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2522, 23, 24syl2anc 696 . . . . . . . . . . 11 (𝜑 → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2625adantr 472 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
274adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
286adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
297adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
30 simpr 479 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
3127, 28, 29, 30reprf 30999 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
32 ffn 6206 . . . . . . . . . . . . 13 (𝑑:(0..^𝑆)⟶𝐴𝑑 Fn (0..^𝑆))
3331, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
3433biantrurd 530 . . . . . . . . . . 11 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)))
35 ffnfv 6551 . . . . . . . . . . 11 (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3634, 35syl6rbbr 279 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3726, 36bitrd 268 . . . . . . . . 9 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3837notbid 307 . . . . . . . 8 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
39 rexnal 3133 . . . . . . . 8 (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)
4038, 39syl6bbr 278 . . . . . . 7 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4140pm5.32da 676 . . . . . 6 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
4218, 41bitr3d 270 . . . . 5 (𝜑 → ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
43 fveq1 6351 . . . . . . . . . 10 (𝑐 = 𝑑 → (𝑐𝑥) = (𝑑𝑥))
4443eleq1d 2824 . . . . . . . . 9 (𝑐 = 𝑑 → ((𝑐𝑥) ∈ 𝐵 ↔ (𝑑𝑥) ∈ 𝐵))
4544notbid 307 . . . . . . . 8 (𝑐 = 𝑑 → (¬ (𝑐𝑥) ∈ 𝐵 ↔ ¬ (𝑑𝑥) ∈ 𝐵))
4645elrab 3504 . . . . . . 7 (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
4746rexbii 3179 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
48 r19.42v 3230 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4947, 48bitri 264 . . . . 5 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
5042, 49syl6bbr 278 . . . 4 (𝜑 → ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ ∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}))
51 rabid 3254 . . . 4 (𝑑 ∈ {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ (𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
52 eliun 4676 . . . 4 (𝑑 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
5350, 51, 523bitr4g 303 . . 3 (𝜑 → (𝑑 ∈ {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ 𝑑 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}))
541, 2, 3, 53eqrd 3763 . 2 (𝜑 → {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
5521, 6, 7reprval 30997 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
568, 55difeq12d 3872 . . 3 (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = ({𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ∖ {𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
57 difrab2 29646 . . 3 ({𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ∖ {𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}) = {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}
5856, 57syl6eq 2810 . 2 (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
59 reprdifc.c . . . 4 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}
6059a1i 11 . . 3 (𝜑𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6160iuneq2d 4699 . 2 (𝜑 𝑥 ∈ (0..^𝑆)𝐶 = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6254, 58, 613eqtr4d 2804 1 (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = 𝑥 ∈ (0..^𝑆)𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  cdif 3712  wss 3715   ciun 4672   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  0cc0 10128  cn 11212  0cn0 11484  cz 11569  ..^cfzo 12659  Σcsu 14615  reprcrepr 30995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-n0 11485  df-z 11570  df-seq 12996  df-sum 14616  df-repr 30996
This theorem is referenced by:  hgt750lema  31044
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