Theorem bj-finsumval0 37308

Description: Value of a finite sum. (Contributed by BJ, 9-Jun-2019.) (Proof shortened by AV, 5-May-2021.)

Hypotheses

Ref	Expression
bj-finsumval0.1	⊢ (𝜑 → 𝐴 ∈ CMnd)
bj-finsumval0.2	⊢ (𝜑 → 𝐼 ∈ Fin)
bj-finsumval0.3	⊢ (𝜑 → 𝐵:𝐼⟶(Base‘𝐴))

Assertion

Ref	Expression
bj-finsumval0	⊢ (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))

Distinct variable groups:   𝐴,𝑠,𝑓,𝑚,𝑛   𝐵,𝑓,𝑚,𝑛,𝑠   𝑓,𝐼,𝑛   𝜑,𝑓,𝑚,𝑠

Allowed substitution hints:   𝜑(𝑛)   𝐼(𝑚,𝑠)

Proof of Theorem bj-finsumval0

Dummy variables 𝑡 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7413	. 2 ⊢ (𝐴 FinSum 𝐵) = ( FinSum ‘⟨𝐴, 𝐵⟩)
2		df-bj-finsum 37307	. . 3 ⊢ FinSum = (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↦ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
3		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → 𝑥 = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘𝑥) = (1st ‘⟨𝐴, 𝐵⟩))
5		bj-finsumval0.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ CMnd)
6	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → 𝐴 ∈ CMnd)
7		bj-finsumval0.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵:𝐼⟶(Base‘𝐴))
8		bj-finsumval0.2	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ Fin)
9	7, 8	fexd 7224	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → 𝐵 ∈ V)
11		op1stg 8005	. . . . . . . . . 10 ⊢ ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
12	6, 10, 11	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
13	4, 12	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (1st ‘𝑥) = 𝐴)
14	3	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 𝐵⟩))
15		op2ndg 8006	. . . . . . . . . 10 ⊢ ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
16	6, 10, 15	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
17	14, 16	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (2nd ‘𝑥) = 𝐵)
18	17	dmeqd 5890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd ‘𝑥) = dom 𝐵)
19	7	fdmd 6721	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐵 = 𝐼)
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → dom 𝐵 = 𝐼)
21	18, 20	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → dom (2nd ‘𝑥) = 𝐼)
22		f1oeq3 6813	. . . . . . . . . . . . . . 15 ⊢ (dom (2nd ‘𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ↔ 𝑓:(1...𝑚)–1-1-onto→𝐼))
23	22	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (dom (2nd ‘𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) → 𝑓:(1...𝑚)–1-1-onto→𝐼))
24	23	ad2antll 729	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) → 𝑓:(1...𝑚)–1-1-onto→𝐼))
25	24	adantrd 491	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) → 𝑓:(1...𝑚)–1-1-onto→𝐼))
26	25	adantr 480	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) → 𝑓:(1...𝑚)–1-1-onto→𝐼))
27		eqidd 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
28		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) → (1st ‘𝑥) = 𝐴)
29	28	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) → (+g‘(1st ‘𝑥)) = (+g‘𝐴))
30	29	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘(1st ‘𝑥)) = (+g‘𝐴))
31		simprrl 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) → (2nd ‘𝑥) = 𝐵)
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (2nd ‘𝑥) = 𝐵)
33	32	fveq1d 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → ((2nd ‘𝑥)‘(𝑓‘𝑛)) = (𝐵‘(𝑓‘𝑛)))
34	33	mpteq2dva 5219	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) → (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))
35	34	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))
36	27, 30, 35	seqeq123d 14033	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))) = seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛)))))
37		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → dom (2nd ‘𝑥) = 𝐼)
38	37	anim1ci 616	. . . . . . . . . . . . . . . . 17 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼))
39		hashfz1 14369	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ0 → (♯‘(1...𝑚)) = 𝑚)
40	39	eqcomd 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 ∈ ℕ0 → 𝑚 = (♯‘(1...𝑚)))
41	40	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → 𝑚 = (♯‘(1...𝑚)))
42		fzfid 13996	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → (1...𝑚) ∈ Fin)
43		19.8a 2182	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥))
44	43	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → ∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥))
45		hasheqf1oi 14374	. . . . . . . . . . . . . . . . . . 19 ⊢ ((1...𝑚) ∈ Fin → (∃𝑓 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) → (♯‘(1...𝑚)) = (♯‘dom (2nd ‘𝑥))))
46	42, 44, 45	sylc 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → (♯‘(1...𝑚)) = (♯‘dom (2nd ‘𝑥)))
47		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → dom (2nd ‘𝑥) = 𝐼)
48	47	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → (♯‘dom (2nd ‘𝑥)) = (♯‘𝐼))
49	41, 46, 48	3eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (𝑚 ∈ ℕ0 ∧ dom (2nd ‘𝑥) = 𝐼)) → 𝑚 = (♯‘𝐼))
50	38, 49	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
51	36, 50	fveq12d 6888	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚) = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))
52	51	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚) ↔ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))
53	52	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚) → 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))
54	53	impancom 451	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) → ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))
55	54	com12 32	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) → 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))
56	26, 55	jcad 512	. . . . . . . . . 10 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) → (𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
57	22	biimprd 248	. . . . . . . . . . . . . 14 ⊢ (dom (2nd ‘𝑥) = 𝐼 → (𝑓:(1...𝑚)–1-1-onto→𝐼 → 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)))
58	57	ad2antll 729	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → (𝑓:(1...𝑚)–1-1-onto→𝐼 → 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)))
59	58	adantr 480	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → (𝑓:(1...𝑚)–1-1-onto→𝐼 → 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)))
60	59	adantrd 491	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥)))
61		eqidd 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 1 = 1)
62		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → (1st ‘𝑥) = 𝐴)
63		tru 1544	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⊤
64	62, 63	jctir 520	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → ((1st ‘𝑥) = 𝐴 ∧ ⊤))
65	64	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → ((1st ‘𝑥) = 𝐴 ∧ ⊤))
66		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ⊤) → (1st ‘𝑥) = 𝐴)
67	66	eqcomd 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ⊤) → 𝐴 = (1st ‘𝑥))
68	65, 67	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝐴 = (1st ‘𝑥))
69	68	fveq2d 6885	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (+g‘𝐴) = (+g‘(1st ‘𝑥)))
70		simpl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼) → (2nd ‘𝑥) = 𝐵)
71	70	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼) → 𝐵 = (2nd ‘𝑥))
72	71	ad2antll 729	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) → 𝐵 = (2nd ‘𝑥))
73	72	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → 𝐵 = (2nd ‘𝑥))
74	73	fveq1d 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ ((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼))) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓‘𝑛)) = ((2nd ‘𝑥)‘(𝑓‘𝑛)))
75	74	adantlrr 721	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) ∧ 𝑛 ∈ ℕ) → (𝐵‘(𝑓‘𝑛)) = ((2nd ‘𝑥)‘(𝑓‘𝑛)))
76	75	mpteq2dva 5219	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))) = (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))
77	61, 69, 76	seqeq123d 14033	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛)))) = seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛)))))
78	59	impcom 407	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥))
79		simprr 772	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 ∈ ℕ0)
80	37	ad2antrl 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → dom (2nd ‘𝑥) = 𝐼)
81	78, 79, 80, 49	syl12anc 836	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → 𝑚 = (♯‘𝐼))
82	81	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (♯‘𝐼) = 𝑚)
83	77, 82	fveq12d 6888	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)) = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))
84	83	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)) ↔ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))
85	84	biimpd 229	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0)) → (𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)) → 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))
86	85	impancom 451	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))) → ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))
87	86	com12 32	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))) → 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)))
88	60, 87	jcad 512	. . . . . . . . . 10 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))) → (𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))))
89	56, 88	impbid 212	. . . . . . . . 9 ⊢ ((((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
90	89	ex 412	. . . . . . . 8 ⊢ (((1st ‘𝑥) = 𝐴 ∧ ((2nd ‘𝑥) = 𝐵 ∧ dom (2nd ‘𝑥) = 𝐼)) → (𝑚 ∈ ℕ0 → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))))
91	13, 17, 21, 90	syl12anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (𝑚 ∈ ℕ0 → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼))))))
92	91	imp 406	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) ∧ 𝑚 ∈ ℕ0) → ((𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
93	92	exbidv 1921	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) ∧ 𝑚 ∈ ℕ0) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
94	93	rexbidva 3163	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚)) ↔ ∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
95	94	iotabidv 6520	. . 3 ⊢ ((𝜑 ∧ 𝑥 = ⟨𝐴, 𝐵⟩) → (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→dom (2nd ‘𝑥) ∧ 𝑠 = (seq1((+g‘(1st ‘𝑥)), (𝑛 ∈ ℕ ↦ ((2nd ‘𝑥)‘(𝑓‘𝑛))))‘𝑚))) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
96		eleq1 2823	. . . . . . . . . 10 ⊢ (𝑡 = 𝐼 → (𝑡 ∈ Fin ↔ 𝐼 ∈ Fin))
97		feq2 6692	. . . . . . . . . 10 ⊢ (𝑡 = 𝐼 → (𝐵:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝐼⟶(Base‘𝐴)))
98	96, 97	anbi12d 632	. . . . . . . . 9 ⊢ (𝑡 = 𝐼 → ((𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
99	98	ceqsexgv 3638	. . . . . . . 8 ⊢ (𝐼 ∈ Fin → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
100	8, 99	syl 17	. . . . . . 7 ⊢ (𝜑 → (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) ↔ (𝐼 ∈ Fin ∧ 𝐵:𝐼⟶(Base‘𝐴))))
101	8, 7, 100	mpbir2and 713	. . . . . 6 ⊢ (𝜑 → ∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))))
102		exsimpr 1869	. . . . . 6 ⊢ (∃𝑡(𝑡 = 𝐼 ∧ (𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴))) → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
103	101, 102	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
104		df-rex 3062	. . . . 5 ⊢ (∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴) ↔ ∃𝑡(𝑡 ∈ Fin ∧ 𝐵:𝑡⟶(Base‘𝐴)))
105	103, 104	sylibr 234	. . . 4 ⊢ (𝜑 → ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))
106		eleq1 2823	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ CMnd ↔ 𝐴 ∈ CMnd))
107		fveq2 6881	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (Base‘𝑦) = (Base‘𝐴))
108	107	feq3d 6698	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑧:𝑡⟶(Base‘𝑦) ↔ 𝑧:𝑡⟶(Base‘𝐴)))
109	108	rexbidv 3165	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦) ↔ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)))
110	106, 109	anbi12d 632	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴))))
111		feq1 6691	. . . . . . . 8 ⊢ (𝑧 = 𝐵 → (𝑧:𝑡⟶(Base‘𝐴) ↔ 𝐵:𝑡⟶(Base‘𝐴)))
112	111	rexbidv 3165	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴) ↔ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴)))
113	112	anbi2d 630	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝐴)) ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
114	110, 113	opelopabg 5518	. . . . 5 ⊢ ((𝐴 ∈ CMnd ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
115	5, 9, 114	syl2anc 584	. . . 4 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))} ↔ (𝐴 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝐵:𝑡⟶(Base‘𝐴))))
116	5, 105, 115	mpbir2and 713	. . 3 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ CMnd ∧ ∃𝑡 ∈ Fin 𝑧:𝑡⟶(Base‘𝑦))})
117		iotaex 6509	. . . 4 ⊢ (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))) ∈ V
118	117	a1i 11	. . 3 ⊢ (𝜑 → (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))) ∈ V)
119	2, 95, 116, 118	fvmptd2 6999	. 2 ⊢ (𝜑 → ( FinSum ‘⟨𝐴, 𝐵⟩) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))
120	1, 119	eqtrid 2783	1 ⊢ (𝜑 → (𝐴 FinSum 𝐵) = (℩𝑠∃𝑚 ∈ ℕ0 ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐼 ∧ 𝑠 = (seq1((+g‘𝐴), (𝑛 ∈ ℕ ↦ (𝐵‘(𝑓‘𝑛))))‘(♯‘𝐼)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541  ∃wex 1779   ∈ wcel 2109  ∃wrex 3061  Vcvv 3464  ⟨cop 4612  {copab 5186   ↦ cmpt 5206  dom cdm 5659  ℩cio 6487  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  Fincfn 8964  1c1 11135  ℕcn 12245  ℕ₀cn0 12506  ...cfz 13529  seqcseq 14024  ♯chash 14353  Basecbs 17233  +_gcplusg 17276  CMndccmn 19766   FinSum cfinsum 37306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-seq 14025  df-hash 14354  df-bj-finsum 37307

This theorem is referenced by: (None)