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Theorem summolem3 15909

Description: Lemma for summo 15912. (Contributed by Mario Carneiro, 29-Mar-2014.)

**Hypotheses**
Ref	Expression
summo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
summo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summo.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
summolem3.4	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
summolem3.5	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
summolem3.6	⊢ (𝜑 → 𝑓:(1...𝑀)⟶1-1onto𝐴)
summolem3.7	⊢ (𝜑 → 𝐾:(1...𝑁)⟶1-1onto𝐴)

**Assertion**
Ref	Expression
summolem3	⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Distinct variable groups: 𝑓,𝑘,𝑛,𝐴 𝑓,𝐹,𝑘,𝑛 𝑘,𝐺,𝑛 𝑘,𝐾,𝑛 𝑘,𝑁,𝑛 𝜑,𝑘,𝑛 𝐵,𝑓,𝑛 𝑘,𝑀,𝑛 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑘) 𝐺(𝑓) 𝐻(𝑓, 𝑘, 𝑛) 𝐾(𝑓) 𝑀(𝑓) 𝑁(𝑓)

Proof of Theorem summolem3 Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 11309	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 11523	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
4	3	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5		addass 11314	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
6	5	adantl 487	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7		summolem3.5	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 500	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 13031	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2874	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		ssidd 3936	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		summolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶1-1onto𝐴)
13		f1ocnv 6859	. . . . . 6 ⊢ (𝑓:(1...𝑀)⟶1-1onto𝐴 → ^◡𝑓:𝐴⟶1-1onto(1...𝑀))
14	12, 13	syl 18	. . . . 5 ⊢ (𝜑 → ^◡𝑓:𝐴⟶1-1onto(1...𝑀))
15		summolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶1-1onto𝐴)
16		f1oco 6870	. . . . 5 ⊢ ((^◡𝑓:𝐴⟶1-1onto(1...𝑀) ∧ 𝐾:(1...𝑁)⟶1-1onto𝐴) → (^◡𝑓 ∘ 𝐾):(1...𝑁)⟶1-1onto(1...𝑀))
17	14, 15, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)⟶1-1onto(1...𝑀))
18		ovex 7476	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 9027	. . . . . . . . 9 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑁)⟶1-1onto(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
20	17, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ≈ (1...𝑀))
21		fzfi 14142	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 14142	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14518	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 705	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 237	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27		nnnn0 12640	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
28		hashfz1 14517	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
29	26, 27, 28	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30		nnnn0 12640	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
31		hashfz1 14517	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
32	8, 30, 31	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2807	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7459	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6842	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)⟶1-1onto(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)⟶1-1onto(1...𝑀)))
36	17, 35	mpbird 260	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶1-1onto(1...𝑀))
37		summo.3	. . . . 5 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
38		fveq2 6908	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
39	38	csbeq1d 3855	. . . . 5 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13714	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6846	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)⟶1-1onto𝐴 → 𝑓:(1...𝑀)⟶𝐴)
43	12, 42	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
44	43	ffvelcdmda 7108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		summo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3158	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3875	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2942	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3865	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2849	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3568	. . . . . 6 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
53	44, 47, 52	sylc 66	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
54	37, 39, 41, 53	fvmptd3 7040	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2864	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6842	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)⟶1-1onto𝐴 ↔ 𝐾:(1...𝑁)⟶1-1onto𝐴))
57	15, 56	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶1-1onto𝐴)
58		f1of 6846	. . . . . . . . . 10 ⊢ (𝐾:(1...𝑀)⟶1-1onto𝐴 → 𝐾:(1...𝑀)⟶𝐴)
59	57, 58	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
60		fvco3 7008	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6912	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)⟶1-1onto𝐴)
64	59	ffvelcdmda 7108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7308	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)⟶1-1onto𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtr2d 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
68	67	csbeq1d 3855	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
69	68	fveq2d 6912	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
70		elfznn 13714	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
71	70	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
72		fveq2 6908	. . . . . . 7 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
73	72	csbeq1d 3855	. . . . . 6 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
74		summolem3.4	. . . . . 6 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
75	73, 74	fvmpti 7015	. . . . 5 ⊢ (𝑖 ∈ ℕ → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
76	71, 75	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
77		f1of 6846	. . . . . . 7 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑀)⟶1-1onto(1...𝑀) → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
78	36, 77	syl 18	. . . . . 6 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
79	78	ffvelcdmda 7108	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
80		elfznn 13714	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
81		fveq2 6908	. . . . . . 7 ⊢ (𝑛 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
82	81	csbeq1d 3855	. . . . . 6 ⊢ (𝑛 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
83	82, 37	fvmpti 7015	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
84	79, 80, 83	3syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
85	69, 76, 84	3eqtr4d 2808	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 14213	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
87	33	fveq2d 6912	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2800	1 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⦋csb 3851 ifcif 4485 class class class wbr 5106 ↦ cmpt 5189 I cid 5549 ^◡ccnv 5654 ∘ ccom 5659 ⟶wf 6554 ⟶1-1ontowf1o 6557 ‘cfv 6558 (class class class)co 7443 ≈ cen 8998 Fincfn 9001 ℂcc 11225 0cc0 11227 1c1 11228 + caddc 11230 ℕcn 12362 ℕ₀cn0 12633 ℤcz 12720 ℤ_≥cuz 12992 ...cfz 13666 seqcseq 14171 ♯chash 14501

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-card 10047 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-n0 12634 df-z 12721 df-uz 12993 df-fz 13667 df-fzo 13816 df-seq 14172 df-hash 14502

This theorem is used by: summolem2a 15910 summo 15912

W3C validator