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Theorem fisumss 11538

Description: Change the index set to a subset in a finite sum. (Contributed by Mario Carneiro, 21-Apr-2014.) (Revised by Jim Kingdon, 23-Sep-2022.)

**Hypotheses**
Ref	Expression
fsumss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
fsumss.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
fsumss.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
fisumss.adc	⊢ (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
fsumss.4	⊢ (𝜑 → 𝐵 ∈ Fin)

**Assertion**
Ref	Expression
fisumss	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Distinct variable groups: 𝐴,𝑗,𝑘 𝐵,𝑗,𝑘 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑗) 𝐶(𝑗,𝑘)

Proof of Theorem fisumss Dummy variables 𝑓 𝑢 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsumss.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sseq0 3489	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
3	1, 2	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 = ∅)
4	3	sumeq1d 11512	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 = ∅)
6	5	sumeq1d 11512	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
7	4, 6	eqtr4d 2229	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
8	7	ex 115	. 2 ⊢ (𝜑 → (𝐵 = ∅ → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
9		cnvimass 5029	. . . . . . . . 9 ⊢ (^◡𝑓 “ 𝐴) ⊆ dom 𝑓
10		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)
11		f1of 5501	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))⟶𝐵)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
13	9, 12	fssdm 5419	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (^◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵)))
14	12	ffnd 5405	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(♯‘𝐵)))
15		elpreima 5678	. . . . . . . . . . . 12 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
16	14, 15	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
17	12	ffvelcdmda 5694	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵)
18	17	ex 115	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵))
19	18	adantrd 279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
20	16, 19	sylbid 150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (^◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
21	20	imp 124	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (^◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵)
22		fsumss.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
23	22	ex 115	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
24	23	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
25		eldif 3163	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
26		fsumss.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
27		0cn 8013	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
28	26, 27	eqeltrdi 2284	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
29	25, 28	sylan2br 288	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
30	29	expr 375	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
31		eleq1w 2254	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
32	31	dcbid 839	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
33		fisumss.adc	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
34	33	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
35		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
36	32, 34, 35	rspcdva 2870	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ 𝐴)
37		exmiddc 837	. . . . . . . . . . . . . 14 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
38	36, 37	syl 14	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
39	24, 30, 38	mpjaod 719	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
40	39	fmpttd 5714	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
41	40	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
42	41	ffvelcdmda 5694	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
43	21, 42	syldan 282	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (^◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
44		eldifi 3282	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
45	44, 17	sylan2 286	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵)
46		eldifn 3283	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (^◡𝑓 “ 𝐴))
47	46	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (^◡𝑓 “ 𝐴))
48	16	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑛 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
49	44	adantl 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
50	49	biantrurd 305	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑓‘𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴)))
51	48, 50	bitr4d 191	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑛 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴))
52	47, 51	mtbid 673	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴)
53	45, 52	eldifd 3164	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴))
54		difss 3286	. . . . . . . . . . . . 13 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
55		resmpt 4991	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))
56	54, 55	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)
57	56	fveq1i 5556	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛))
58		fvres 5579	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
59	57, 58	eqtr3id 2240	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
60	53, 59	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
61		c0ex 8015	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
62	61	elsn2 3653	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ {0} ↔ 𝐶 = 0)
63	26, 62	sylibr 134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {0})
64	63	fmpttd 5714	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0})
65	64	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0})
66	65, 53	ffvelcdmd 5695	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0})
67		elsni 3637	. . . . . . . . . 10 ⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 0)
68	66, 67	syl 14	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 0)
69	60, 68	eqtr3d 2228	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 0)
70		eleq1 2256	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑓‘𝑢) → (𝑗 ∈ 𝐴 ↔ (𝑓‘𝑢) ∈ 𝐴))
71	70	dcbid 839	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑓‘𝑢) → (DECID 𝑗 ∈ 𝐴 ↔ DECID (𝑓‘𝑢) ∈ 𝐴))
72	33	ad3antrrr 492	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
73	12	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
74		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ (1...(♯‘𝐵)))
75	73, 74	ffvelcdmd 5695	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑢) ∈ 𝐵)
76	71, 72, 75	rspcdva 2870	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID (𝑓‘𝑢) ∈ 𝐴)
77	10	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)
78		f1ofun 5503	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Fun 𝑓)
79	77, 78	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → Fun 𝑓)
80		f1odm 5505	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → dom 𝑓 = (1...(♯‘𝐵)))
81	77, 80	syl 14	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → dom 𝑓 = (1...(♯‘𝐵)))
82	74, 81	eleqtrrd 2273	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ dom 𝑓)
83		fvimacnv 5674	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ 𝑢 ∈ dom 𝑓) → ((𝑓‘𝑢) ∈ 𝐴 ↔ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
84	79, 82, 83	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → ((𝑓‘𝑢) ∈ 𝐴 ↔ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
85	84	dcbid 839	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → (DECID (𝑓‘𝑢) ∈ 𝐴 ↔ DECID 𝑢 ∈ (^◡𝑓 “ 𝐴)))
86	76, 85	mpbid 147	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
87		elpreima 5678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑢) ∈ 𝐴)))
88		simpl 109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑢) ∈ 𝐴) → 𝑢 ∈ (1...(♯‘𝐵)))
89	87, 88	biimtrdi 163	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (^◡𝑓 “ 𝐴) → 𝑢 ∈ (1...(♯‘𝐵))))
90	89	con3d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
91	14, 90	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
92	91	adantr 276	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
93	92	imp 124	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴))
94	93	olcd 735	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → (𝑢 ∈ (^◡𝑓 “ 𝐴) ∨ ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
95		df-dc 836	. . . . . . . . . . 11 ⊢ (DECID 𝑢 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑢 ∈ (^◡𝑓 “ 𝐴) ∨ ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
96	94, 95	sylibr 134	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
97		eluzelz 9604	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℤ)
98	97	adantl 277	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℤ)
99		1zzd 9347	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 1 ∈ ℤ)
100		simplrl 535	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐵) ∈ ℕ)
101	100	nnzd 9441	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐵) ∈ ℤ)
102		fzdcel 10109	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℤ ∧ 1 ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
103	98, 99, 101, 102	syl3anc 1249	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
104		exmiddc 837	. . . . . . . . . . 11 ⊢ (DECID 𝑢 ∈ (1...(♯‘𝐵)) → (𝑢 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑢 ∈ (1...(♯‘𝐵))))
105	103, 104	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (𝑢 ∈ (1...(♯‘𝐵)) ∨ ¬ 𝑢 ∈ (1...(♯‘𝐵))))
106	86, 96, 105	mpjaodan 799	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
107	106	ralrimiva 2567	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∀𝑢 ∈ (ℤ_≥‘1)DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
108		1zzd 9347	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 1 ∈ ℤ)
109		fzssuz 10134	. . . . . . . . 9 ⊢ (1...(♯‘𝐵)) ⊆ (ℤ_≥‘1)
110	109	a1i 9	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ⊆ (ℤ_≥‘1))
111	103	ralrimiva 2567	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∀𝑢 ∈ (ℤ_≥‘1)DECID 𝑢 ∈ (1...(♯‘𝐵)))
112	13, 43, 69, 107, 108, 110, 111	isumss 11537	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑛 ∈ (^◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
113	1	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
114	113	resmptd 4994	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶))
115	114	fveq1d 5557	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
116		fvres 5579	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
117	116	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
118	115, 117	eqtr3d 2228	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
119	118	sumeq2dv 11514	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
120		fveq2 5555	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
121	1	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵)
122		fsumss.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ Fin)
123		ssfidc 6993	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴) → 𝐴 ∈ Fin)
124	122, 1, 33, 123	syl3anc 1249	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
125	124	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ∈ Fin)
126	121, 10, 125	preimaf1ofi 7012	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (^◡𝑓 “ 𝐴) ∈ Fin)
127		f1of1 5500	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
128	10, 127	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1→𝐵)
129		f1ores 5516	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘𝐵))–1-1→𝐵 ∧ (^◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (^◡𝑓 “ 𝐴)))
130	128, 13, 129	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (^◡𝑓 “ 𝐴)))
131		f1ofo 5508	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
132	10, 131	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–onto→𝐵)
133		foimacnv 5519	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (^◡𝑓 “ 𝐴)) = 𝐴)
134	132, 121, 133	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (^◡𝑓 “ 𝐴)) = 𝐴)
135		f1oeq3 5491	. . . . . . . . . . 11 ⊢ ((𝑓 “ (^◡𝑓 “ 𝐴)) = 𝐴 → ((𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (^◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→𝐴))
136	134, 135	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (^◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→𝐴))
137	130, 136	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→𝐴)
138		fvres 5579	. . . . . . . . . 10 ⊢ (𝑛 ∈ (^◡𝑓 “ 𝐴) → ((𝑓 ↾ (^◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
139	138	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (^◡𝑓 “ 𝐴)) → ((𝑓 ↾ (^◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
140	121	sselda 3180	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
141	41	ffvelcdmda 5694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
142	140, 141	syldan 282	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
143	120, 126, 137, 139, 142	fsumf1o 11536	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (^◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
144	119, 143	eqtrd 2226	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (^◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
145		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℕ)
146	145	nnzd 9441	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℤ)
147	108, 146	fzfigd 10505	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
148		eqidd 2194	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛))
149	120, 147, 10, 148, 141	fsumf1o 11536	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (1...(♯‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
150	112, 144, 149	3eqtr4d 2236	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
151	22	ralrimiva 2567	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
152		sumfct 11520	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
153	151, 152	syl 14	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
154	153	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
155	22	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
156		simpll 527	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝜑)
157		simplr 528	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐵)
158		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐴)
159	157, 158	eldifd 3164	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐵 ∖ 𝐴))
160	156, 159, 26	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 = 0)
161		0cnd 8014	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 0 ∈ ℂ)
162	160, 161	eqeltrd 2270	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
163	155, 162, 38	mpjaodan 799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
164	163	ralrimiva 2567	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
165		sumfct 11520	. . . . . . . 8 ⊢ (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶)
166	164, 165	syl 14	. . . . . . 7 ⊢ (𝜑 → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶)
167	166	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶)
168	150, 154, 167	3eqtr3d 2234	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
169	168	expr 375	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
170	169	exlimdv 1830	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
171	170	expimpd 363	. 2 ⊢ (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
172		fz1f1o 11521	. . 3 ⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
173	122, 172	syl 14	. 2 ⊢ (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)))
174	8, 171, 173	mpjaod 719	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 = wceq 1364 ∃wex 1503 ∈ wcel 2164 ∀wral 2472 ∖ cdif 3151 ⊆ wss 3154 ∅c0 3447 {csn 3619 ↦ cmpt 4091 ^◡ccnv 4659 dom cdm 4660 ↾ cres 4662 “ cima 4663 Fun wfun 5249 Fn wfn 5250 ⟶wf 5251 –1-1→wf1 5252 –onto→wfo 5253 –1-1-onto→wf1o 5254 ‘cfv 5255 (class class class)co 5919 Fincfn 6796 ℂcc 7872 0cc0 7874 1c1 7875 ℕcn 8984 ℤcz 9320 ℤ_≥cuz 9595 ...cfz 10077 ♯chash 10849 Σcsu 11499

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-frec 6446 df-1o 6471 df-oadd 6475 df-er 6589 df-en 6797 df-dom 6798 df-fin 6799 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-seqfrec 10522 df-exp 10613 df-ihash 10850 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-clim 11425 df-sumdc 11500

This theorem is referenced by: isumss2 11539 ply1termlem 14921 plyaddlem1 14926 plymullem1 14927 dvply1 14943

W3C validator