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Theorem reprdifc 31087
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 2009 . . 3 𝑑𝜑
2 nfrab1 3269 . . 3 𝑑{𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}
3 nfcv 2906 . . 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 11726 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7 reprdifc.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ0)
84, 6, 7reprval 31070 . . . . . . . . . 10 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
98eleq2d 2829 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
10 rabid 3262 . . . . . . . . 9 (𝑑 ∈ {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
119, 10syl6bb 278 . . . . . . . 8 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀)))
1211anbi1d 623 . . . . . . 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 3741 . . . . . . . . . 10 (𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))))
1413anbi1i 617 . . . . . . . . 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 636 . . . . . . . . 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 266 . . . . . . . 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 273 . . . . . 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 11280 . . . . . . . . . . . . . 14 ℕ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
21 reprdifc.b . . . . . . . . . . . . 13 (𝜑𝐵 ⊆ ℕ)
2220, 21ssexd 4965 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
23 ovexd 6875 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ∈ V)
24 elmapg 8072 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2522, 23, 24syl2anc 579 . . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
3127, 28, 29, 30reprf 31072 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
3231ffnd 6223 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
3332biantrurd 528 . . . . . . . . . . 11 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)))
34 ffnfv 6577 . . . . . . . . . . 11 (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3533, 34syl6rbbr 281 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3626, 35bitrd 270 . . . . . . . . 9 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3736notbid 309 . . . . . . . 8 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
38 rexnal 3140 . . . . . . . 8 (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)
3937, 38syl6bbr 280 . . . . . . 7 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4039pm5.32da 574 . . . . . 6 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
4118, 40bitr3d 272 . . . . 5 (𝜑 → ((𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
42 fveq1 6373 . . . . . . . . . 10 (𝑐 = 𝑑 → (𝑐𝑥) = (𝑑𝑥))
4342eleq1d 2828 . . . . . . . . 9 (𝑐 = 𝑑 → ((𝑐𝑥) ∈ 𝐵 ↔ (𝑑𝑥) ∈ 𝐵))
4443notbid 309 . . . . . . . 8 (𝑐 = 𝑑 → (¬ (𝑐𝑥) ∈ 𝐵 ↔ ¬ (𝑑𝑥) ∈ 𝐵))
4544elrab 3518 . . . . . . 7 (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
4645rexbii 3187 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
47 r19.42v 3238 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4846, 47bitri 266 . . . . 5 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4941, 48syl6bbr 280 . . . 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}))
50 rabid 3262 . . . 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))
51 eliun 4679 . . . 4 (𝑑 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ ∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
5249, 50, 513bitr4g 305 . . 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}))
531, 2, 3, 52eqrd 3779 . 2 (𝜑 → {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
5421, 6, 7reprval 31070 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
558, 54difeq12d 3890 . . 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}))
56 difrab2 29722 . . 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}
5755, 56syl6eq 2814 . 2 (𝜑 → ((𝐴(repr‘𝑆)𝑀) ∖ (𝐵(repr‘𝑆)𝑀)) = {𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
58 reprdifc.c . . . 4 𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵}
5958a1i 11 . . 3 (𝜑𝐶 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6059iuneq2d 4702 . 2 (𝜑 𝑥 ∈ (0..^𝑆)𝐶 = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6153, 57, 603eqtr4d 2808 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 197  wa 384   = wceq 1652  wcel 2155  wral 3054  wrex 3055  {crab 3058  Vcvv 3349  cdif 3728  wss 3731   ciun 4675   Fn wfn 6062  wf 6063  cfv 6067  (class class class)co 6841  𝑚 cmap 8059  0cc0 10188  cn 11273  0cn0 11537  cz 11623  ..^cfzo 12672  Σcsu 14702  reprcrepr 31068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-er 7946  df-map 8061  df-en 8160  df-dom 8161  df-sdom 8162  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-nn 11274  df-n0 11538  df-z 11624  df-seq 13008  df-sum 14703  df-repr 31069
This theorem is referenced by:  hgt750lema  31117
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