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Theorem reprdifc 32012
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 1915 . . 3 𝑑𝜑
2 nfrab1 3340 . . 3 𝑑{𝑑 ∈ ((𝐴m (0..^𝑆)) ∖ (𝐵m (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}
3 nfcv 2958 . . 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 12077 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7 reprdifc.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ0)
84, 6, 7reprval 31995 . . . . . . . . . 10 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
98eleq2d 2878 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
10 rabid 3334 . . . . . . . . 9 (𝑑 ∈ {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
119, 10syl6bb 290 . . . . . . . 8 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀)))
1211anbi1d 632 . . . . . . 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 3894 . . . . . . . . . 10 (𝑑 ∈ ((𝐴m (0..^𝑆)) ∖ (𝐵m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵m (0..^𝑆))))
1413anbi1i 626 . . . . . . . . 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 645 . . . . . . . . 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 278 . . . . . . . 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 285 . . . . . 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 11635 . . . . . . . . . . . . . 14 ℕ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
21 reprdifc.b . . . . . . . . . . . . 13 (𝜑𝐵 ⊆ ℕ)
2220, 21ssexd 5195 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
23 ovexd 7174 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ∈ V)
24 elmapg 8406 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2522, 23, 24syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2625adantr 484 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
274adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
286adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
297adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
30 simpr 488 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
3127, 28, 29, 30reprf 31997 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
3231ffnd 6492 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
3332biantrurd 536 . . . . . . . . . . 11 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)))
34 ffnfv 6863 . . . . . . . . . . 11 (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3533, 34syl6rbbr 293 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3626, 35bitrd 282 . . . . . . . . 9 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3736notbid 321 . . . . . . . 8 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
38 rexnal 3204 . . . . . . . 8 (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)
3937, 38syl6bbr 292 . . . . . . 7 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4039pm5.32da 582 . . . . . 6 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
4118, 40bitr3d 284 . . . . 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 6648 . . . . . . . . . 10 (𝑐 = 𝑑 → (𝑐𝑥) = (𝑑𝑥))
4342eleq1d 2877 . . . . . . . . 9 (𝑐 = 𝑑 → ((𝑐𝑥) ∈ 𝐵 ↔ (𝑑𝑥) ∈ 𝐵))
4443notbid 321 . . . . . . . 8 (𝑐 = 𝑑 → (¬ (𝑐𝑥) ∈ 𝐵 ↔ ¬ (𝑑𝑥) ∈ 𝐵))
4544elrab 3631 . . . . . . 7 (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
4645rexbii 3213 . . . . . 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 3306 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4846, 47bitri 278 . . . . 5 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4941, 48syl6bbr 292 . . . 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 3334 . . . 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 4888 . . . 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 317 . . 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 3937 . 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 31995 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
558, 54difeq12d 4054 . . 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 30272 . . 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, 56eqtrdi 2852 . 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 4913 . 2 (𝜑 𝑥 ∈ (0..^𝑆)𝐶 = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6153, 57, 603eqtr4d 2846 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 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  wss 3884   ciun 4884   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  m cmap 8393  0cc0 10530  cn 11629  0cn0 11889  cz 11973  ..^cfzo 13032  Σcsu 15038  reprcrepr 31993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-i2m1 10598  ax-1ne0 10599  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-map 8395  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-seq 13369  df-sum 15039  df-repr 31994
This theorem is referenced by:  hgt750lema  32042
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