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Theorem fisumss 11047

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 3368	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 = ∅) → 𝐴 = ∅)
3	1, 2	sylan 279	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 = ∅)
4	3	sumeq1d 11021	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
5		simpr 109	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 = ∅)
6	5	sumeq1d 11021	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐵 𝐶 = Σ𝑘 ∈ ∅ 𝐶)
7	4, 6	eqtr4d 2148	. . 3 ⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
8	7	ex 114	. 2 ⊢ (𝜑 → (𝐵 = ∅ → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
9		cnvimass 4858	. . . . . . . . 9 ⊢ (^◡𝑓 “ 𝐴) ⊆ dom 𝑓
10		simprr 504	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)
11		f1of 5321	. . . . . . . . . 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 5243	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (^◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵)))
14	12	ffnd 5229	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(♯‘𝐵)))
15		elpreima 5491	. . . . . . . . . . . 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	ffvelrnda 5507	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵)
18	17	ex 114	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵))
19	18	adantrd 275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
20	16, 19	sylbid 149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (^◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵))
21	20	imp 123	. . . . . . . . 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 114	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
24	23	adantr 272	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
25		eldif 3044	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴))
26		fsumss.3	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0)
27		0cn 7676	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℂ
28	26, 27	syl6eqel 2203	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ)
29	25, 28	sylan2br 284	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ)
30	29	expr 370	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ))
31		eleq1w 2173	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
32	31	dcbid 806	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
33		fisumss.adc	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
34	33	adantr 272	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴)
35		simpr 109	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝑘 ∈ 𝐵)
36	32, 34, 35	rspcdva 2763	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → DECID 𝑘 ∈ 𝐴)
37		exmiddc 804	. . . . . . . . . . . . . 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 690	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
40	39	fmpttd 5527	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
41	40	adantr 272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ)
42	41	ffvelrnda 5507	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
43	21, 42	syldan 278	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (^◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ)
44		eldifi 3162	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵)))
45	44, 17	sylan2 282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵)
46		eldifn 3163	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (^◡𝑓 “ 𝐴))
47	46	adantl 273	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (^◡𝑓 “ 𝐴))
48	16	adantr 272	. . . . . . . . . . . . 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 273	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵)))
50	49	biantrurd 301	. . . . . . . . . . . . 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 190	. . . . . . . . . . . 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 644	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴)
53	45, 52	eldifd 3045	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴))
54		difss 3166	. . . . . . . . . . . . 13 ⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵
55		resmpt 4823	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶))
56	54, 55	ax-mp 7	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)
57	56	fveq1i 5374	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛))
58		fvres 5397	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
59	57, 58	syl5eqr 2159	. . . . . . . . . 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 7678	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
62	61	elsn2 3523	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ {0} ↔ 𝐶 = 0)
63	26, 62	sylibr 133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {0})
64	63	fmpttd 5527	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0})
65	64	ad2antrr 477	. . . . . . . . . . 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	ffvelrnd 5508	. . . . . . . . . 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 3509	. . . . . . . . . 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 2147	. . . . . . . 8 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (^◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 0)
70		eleq1 2175	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝑓‘𝑢) → (𝑗 ∈ 𝐴 ↔ (𝑓‘𝑢) ∈ 𝐴))
71	70	dcbid 806	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝑓‘𝑢) → (DECID 𝑗 ∈ 𝐴 ↔ DECID (𝑓‘𝑢) ∈ 𝐴))
72	33	ad3antrrr 481	. . . . . . . . . . . 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 477	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑓:(1...(♯‘𝐵))⟶𝐵)
74		simpr 109	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ (1...(♯‘𝐵)))
75	73, 74	ffvelrnd 5508	. . . . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID (𝑓‘𝑢) ∈ 𝐴)
77	10	ad2antrr 477	. . . . . . . . . . . . . 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 5323	. . . . . . . . . . . . . 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 5325	. . . . . . . . . . . . . . 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 2192	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → 𝑢 ∈ dom 𝑓)
83		fvimacnv 5487	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑓 ∧ 𝑢 ∈ dom 𝑓) → ((𝑓‘𝑢) ∈ 𝐴 ↔ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
84	79, 82, 83	syl2anc 406	. . . . . . . . . . . 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 806	. . . . . . . . . . 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 146	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
87		elpreima 5491	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑢) ∈ 𝐴)))
88		simpl 108	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑢) ∈ 𝐴) → 𝑢 ∈ (1...(♯‘𝐵)))
89	87, 88	syl6bi 162	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑢 ∈ (^◡𝑓 “ 𝐴) → 𝑢 ∈ (1...(♯‘𝐵))))
90	89	con3d 603	. . . . . . . . . . . . . . 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 272	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (¬ 𝑢 ∈ (1...(♯‘𝐵)) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
93	92	imp 123	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴))
94	93	olcd 706	. . . . . . . . . . 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 803	. . . . . . . . . . 11 ⊢ (DECID 𝑢 ∈ (^◡𝑓 “ 𝐴) ↔ (𝑢 ∈ (^◡𝑓 “ 𝐴) ∨ ¬ 𝑢 ∈ (^◡𝑓 “ 𝐴)))
96	94, 95	sylibr 133	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑢 ∈ (1...(♯‘𝐵))) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
97		eluzelz 9231	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ (ℤ_≥‘1) → 𝑢 ∈ ℤ)
98	97	adantl 273	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 𝑢 ∈ ℤ)
99		1zzd 8979	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → 1 ∈ ℤ)
100		simplrl 507	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐵) ∈ ℕ)
101	100	nnzd 9070	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → (♯‘𝐵) ∈ ℤ)
102		fzdcel 9707	. . . . . . . . . . . 12 ⊢ ((𝑢 ∈ ℤ ∧ 1 ∈ ℤ ∧ (♯‘𝐵) ∈ ℤ) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
103	98, 99, 101, 102	syl3anc 1197	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ∈ (1...(♯‘𝐵)))
104		exmiddc 804	. . . . . . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑢 ∈ (ℤ_≥‘1)) → DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
107	106	ralrimiva 2477	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∀𝑢 ∈ (ℤ_≥‘1)DECID 𝑢 ∈ (^◡𝑓 “ 𝐴))
108		1zzd 8979	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 1 ∈ ℤ)
109		fzssuz 9732	. . . . . . . . 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 2477	. . . . . . . 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 11046	. . . . . . 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 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝐴 ⊆ 𝐵)
114	113	resmptd 4826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶))
115	114	fveq1d 5375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚))
116		fvres 5397	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
117	116	adantl 273	. . . . . . . . . 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 2147	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚))
119	118	sumeq2dv 11023	. . . . . . . 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 5373	. . . . . . . . 9 ⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)))
121	1	adantr 272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵)
122		fsumss.4	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ Fin)
123		ssfidc 6773	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ⊆ 𝐵 ∧ ∀𝑗 ∈ 𝐵 DECID 𝑗 ∈ 𝐴) → 𝐴 ∈ Fin)
124	122, 1, 33, 123	syl3anc 1197	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ Fin)
125	124	adantr 272	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ∈ Fin)
126	121, 10, 125	preimaf1ofi 6789	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (^◡𝑓 “ 𝐴) ∈ Fin)
127		f1of1 5320	. . . . . . . . . . . 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 5336	. . . . . . . . . . 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 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (^◡𝑓 “ 𝐴)))
131		f1ofo 5328	. . . . . . . . . . . . 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 5339	. . . . . . . . . . . 12 ⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (^◡𝑓 “ 𝐴)) = 𝐴)
134	132, 121, 133	syl2anc 406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (^◡𝑓 “ 𝐴)) = 𝐴)
135		f1oeq3 5314	. . . . . . . . . . 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 146	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (^◡𝑓 “ 𝐴)):(^◡𝑓 “ 𝐴)–1-1-onto→𝐴)
138		fvres 5397	. . . . . . . . . 10 ⊢ (𝑛 ∈ (^◡𝑓 “ 𝐴) → ((𝑓 ↾ (^◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
139	138	adantl 273	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (^◡𝑓 “ 𝐴)) → ((𝑓 ↾ (^◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛))
140	121	sselda 3061	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵)
141	41	ffvelrnda 5507	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ)
142	140, 141	syldan 278	. . . . . . . . 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 11045	. . . . . . . 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 2145	. . . . . . 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 503	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℕ)
146	145	nnzd 9070	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈ ℤ)
147	108, 146	fzfigd 10091	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (1...(♯‘𝐵)) ∈ Fin)
148		eqidd 2114	. . . . . . . 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 11045	. . . . . . 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 2155	. . . . . 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 2477	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ ℂ)
152		sumfct 11029	. . . . . . . 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 272	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶)
155	22	adantlr 466	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
156		simpll 501	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝜑)
157		simplr 502	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐵)
158		simpr 109	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → ¬ 𝑘 ∈ 𝐴)
159	157, 158	eldifd 3045	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝑘 ∈ (𝐵 ∖ 𝐴))
160	156, 159, 26	syl2anc 406	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 = 0)
161		0cnd 7677	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 0 ∈ ℂ)
162	160, 161	eqeltrd 2189	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐵) ∧ ¬ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
163	155, 162, 38	mpjaodan 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ)
164	163	ralrimiva 2477	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ)
165		sumfct 11029	. . . . . . . 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 272	. . . . . 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 2153	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)
169	168	expr 370	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
170	169	exlimdv 1771	. . 3 ⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
171	170	expimpd 358	. 2 ⊢ (𝜑 → (((♯‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶))
172		fz1f1o 11030	. . 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 690	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 680 DECID wdc 802 = wceq 1312 ∃wex 1449 ∈ wcel 1461 ∀wral 2388 ∖ cdif 3032 ⊆ wss 3035 ∅c0 3327 {csn 3491 ↦ cmpt 3947 ^◡ccnv 4496 dom cdm 4497 ↾ cres 4499 “ cima 4500 Fun wfun 5073 Fn wfn 5074 ⟶wf 5075 –1-1→wf1 5076 –onto→wfo 5077 –1-1-onto→wf1o 5078 ‘cfv 5079 (class class class)co 5726 Fincfn 6586 ℂcc 7539 0cc0 7541 1c1 7542 ℕcn 8624 ℤcz 8952 ℤ_≥cuz 9222 ...cfz 9677 ♯chash 10408 Σcsu 11008

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 ax-arch 7658 ax-caucvg 7659

This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-isom 5088 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-irdg 6219 df-frec 6240 df-1o 6265 df-oadd 6269 df-er 6381 df-en 6587 df-dom 6588 df-fin 6589 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-n0 8876 df-z 8953 df-uz 9223 df-q 9308 df-rp 9338 df-fz 9678 df-fzo 9807 df-seqfrec 10106 df-exp 10180 df-ihash 10409 df-cj 10501 df-re 10502 df-im 10503 df-rsqrt 10656 df-abs 10657 df-clim 10934 df-sumdc 11009

This theorem is referenced by: isumss2 11048

W3C validator