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Theorem reprdifc 34620
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 1911 . . 3 𝑑𝜑
2 nfrab1 3453 . . 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 12636 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7 reprdifc.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ0)
84, 6, 7reprval 34603 . . . . . . . . . 10 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
98eleq2d 2824 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
10 rabid 3454 . . . . . . . . 9 (𝑑 ∈ {𝑑 ∈ (𝐴m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀))
119, 10bitrdi 287 . . . . . . . 8 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀)))
1211anbi1d 631 . . . . . . 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 3972 . . . . . . . . . 10 (𝑑 ∈ ((𝐴m (0..^𝑆)) ∖ (𝐵m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴m (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵m (0..^𝑆))))
1413anbi1i 624 . . . . . . . . 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 646 . . . . . . . . 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 275 . . . . . . . 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 282 . . . . . 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 12269 . . . . . . . . . . . . . 14 ℕ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
21 reprdifc.b . . . . . . . . . . . . 13 (𝜑𝐵 ⊆ ℕ)
2220, 21ssexd 5329 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
23 ovexd 7465 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ∈ V)
24 elmapg 8877 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2522, 23, 24syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2625adantr 480 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
27 ffnfv 7138 . . . . . . . . . . 11 (𝑑:(0..^𝑆)⟶𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
284adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
296adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
307adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
31 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
3228, 29, 30, 31reprf 34605 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
3332ffnd 6737 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
3433biantrurd 532 . . . . . . . . . . 11 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)))
3527, 34bitr4id 290 . . . . . . . . . 10 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑:(0..^𝑆)⟶𝐵 ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3626, 35bitrd 279 . . . . . . . . 9 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
3736notbid 318 . . . . . . . 8 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵))
38 rexnal 3097 . . . . . . . 8 (∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵 ↔ ¬ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)
3937, 38bitr4di 289 . . . . . . 7 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (¬ 𝑑 ∈ (𝐵m (0..^𝑆)) ↔ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4039pm5.32da 579 . . . . . 6 (𝜑 → ((𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ 𝑑 ∈ (𝐵m (0..^𝑆))) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵)))
4118, 40bitr3d 281 . . . . 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 6905 . . . . . . . . . 10 (𝑐 = 𝑑 → (𝑐𝑥) = (𝑑𝑥))
4342eleq1d 2823 . . . . . . . . 9 (𝑐 = 𝑑 → ((𝑐𝑥) ∈ 𝐵 ↔ (𝑑𝑥) ∈ 𝐵))
4443notbid 318 . . . . . . . 8 (𝑐 = 𝑑 → (¬ (𝑐𝑥) ∈ 𝐵 ↔ ¬ (𝑑𝑥) ∈ 𝐵))
4544elrab 3694 . . . . . . 7 (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
4645rexbii 3091 . . . . . 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 3188 . . . . . 6 (∃𝑥 ∈ (0..^𝑆)(𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4846, 47bitri 275 . . . . 5 (∃𝑥 ∈ (0..^𝑆)𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ∃𝑥 ∈ (0..^𝑆) ¬ (𝑑𝑥) ∈ 𝐵))
4941, 48bitr4di 289 . . . 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 3454 . . . 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 4999 . . . 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 314 . . 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 4014 . 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 34603 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
558, 54difeq12d 4136 . . 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 32525 . . 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 2790 . 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 5026 . 2 (𝜑 𝑥 ∈ (0..^𝑆)𝐶 = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6153, 57, 603eqtr4d 2784 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 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  cdif 3959  wss 3962   ciun 4995   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  0cc0 11152  cn 12263  0cn0 12523  cz 12610  ..^cfzo 13690  Σcsu 15718  reprcrepr 34601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-i2m1 11220  ax-1ne0 11221  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-map 8866  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-seq 14039  df-sum 15719  df-repr 34602
This theorem is referenced by:  hgt750lema  34650
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