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Theorem reprdifc 30833
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 1883 . . 3 𝑑𝜑
2 nfrab1 3152 . . 3 𝑑{𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}
3 nfcv 2793 . . 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 11518 . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
7 reprdifc.s . . . . . . . . . . 11 (𝜑𝑆 ∈ ℕ0)
84, 6, 7reprval 30816 . . . . . . . . . 10 (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
98eleq2d 2716 . . . . . . . . 9 (𝜑 → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ↔ 𝑑 ∈ {𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀}))
10 rabid 3145 . . . . . . . . 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 741 . . . . . . 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 3617 . . . . . . . . . 10 (𝑑 ∈ ((𝐴𝑚 (0..^𝑆)) ∖ (𝐵𝑚 (0..^𝑆))) ↔ (𝑑 ∈ (𝐴𝑚 (0..^𝑆)) ∧ ¬ 𝑑 ∈ (𝐵𝑚 (0..^𝑆))))
1413anbi1i 731 . . . . . . . . 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 856 . . . . . . . . 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 11064 . . . . . . . . . . . . . 14 ℕ ∈ V
2019a1i 11 . . . . . . . . . . . . 13 (𝜑 → ℕ ∈ V)
21 reprdifc.b . . . . . . . . . . . . 13 (𝜑𝐵 ⊆ ℕ)
2220, 21ssexd 4838 . . . . . . . . . . . 12 (𝜑𝐵 ∈ V)
23 ovexd 6720 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ∈ V)
24 elmapg 7912 . . . . . . . . . . . 12 ((𝐵 ∈ V ∧ (0..^𝑆) ∈ V) → (𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ↔ 𝑑:(0..^𝑆)⟶𝐵))
2522, 23, 24syl2anc 694 . . . . . . . . . . 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))
274adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
286adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
297adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ0)
30 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
3127, 28, 29, 30reprf 30818 . . . . . . . . . . . . 13 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
32 ffn 6083 . . . . . . . . . . . . 13 (𝑑:(0..^𝑆)⟶𝐴𝑑 Fn (0..^𝑆))
3331, 32syl 17 . . . . . . . . . . . 12 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 Fn (0..^𝑆))
3433biantrurd 528 . . . . . . . . . . 11 ((𝜑𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵 ↔ (𝑑 Fn (0..^𝑆) ∧ ∀𝑥 ∈ (0..^𝑆)(𝑑𝑥) ∈ 𝐵)))
35 ffnfv 6428 . . . . . . . . . . 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 3024 . . . . . . . 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 674 . . . . . 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 6228 . . . . . . . . . 10 (𝑐 = 𝑑 → (𝑐𝑥) = (𝑑𝑥))
4443eleq1d 2715 . . . . . . . . 9 (𝑐 = 𝑑 → ((𝑐𝑥) ∈ 𝐵 ↔ (𝑑𝑥) ∈ 𝐵))
4544notbid 307 . . . . . . . 8 (𝑐 = 𝑑 → (¬ (𝑐𝑥) ∈ 𝐵 ↔ ¬ (𝑑𝑥) ∈ 𝐵))
4645elrab 3396 . . . . . . 7 (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑𝑥) ∈ 𝐵))
4746rexbii 3070 . . . . . 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 3121 . . . . . 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 3145 . . . 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 4556 . . . 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 3655 . 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 30816 . . . 4 (𝜑 → (𝐵(repr‘𝑆)𝑀) = {𝑑 ∈ (𝐵𝑚 (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑𝑎) = 𝑀})
568, 55difeq12d 3762 . . 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 29465 . . 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 2701 . 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 4579 . 2 (𝜑 𝑥 ∈ (0..^𝑆)𝐶 = 𝑥 ∈ (0..^𝑆){𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐𝑥) ∈ 𝐵})
6254, 58, 613eqtr4d 2695 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 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cdif 3604  wss 3607   ciun 4552   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  cn 11058  0cn0 11330  cz 11415  ..^cfzo 12504  Σcsu 14460  reprcrepr 30814
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-om 7108  df-1st 7210  df-2nd 7211  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-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  df-z 11416  df-seq 12842  df-sum 14461  df-repr 30815
This theorem is referenced by:  hgt750lema  30863
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