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Theorem prodmolem3 15986

Description: Lemma for prodmo 15989. (Contributed by Scott Fenton, 4-Dec-2017.)

**Hypotheses**
Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmo.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
prodmolem3.4	⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
prodmolem3.5	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
prodmolem3.6	⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
prodmolem3.7	⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)

**Assertion**
Ref	Expression
prodmolem3	⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

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

Proof of Theorem prodmolem3 Dummy variables 𝑚 𝑖 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 11183	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
2	1	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3		mulcom 11185	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
4	3	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5		mulass 11187	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
6	5	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
7		prodmolem3.5	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 499	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 12900	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2879	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		ssidd 3968	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		prodmolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
13		f1ocnv 6834	. . . . . 6 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
14	12, 13	syl 18	. . . . 5 ⊢ (𝜑 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
15		prodmolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
16		f1oco 6845	. . . . 5 ⊢ ((^◡𝑓:𝐴–1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto→𝐴) → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17	14, 15, 16	syl2anc 595	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18		ovex 7444	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 8968	. . . . . . . . 9 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
20	17, 19	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ≈ (1...𝑀))
21		fzfi 14007	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 14007	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14382	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 704	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 237	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12564	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		hashfz1 14381	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
29	27, 28	syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30	8	nnnn0d 12564	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
31		hashfz1 14381	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
32	30, 31	syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2813	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6817	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
36	17, 35	mpbird 260	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37		prodmo.3	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
38		fveq2 6882	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑓‘𝑗) = (𝑓‘𝑚))
39	38	csbeq1d 3865	. . . . 5 ⊢ (𝑗 = 𝑚 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13580	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6821	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
43	12, 42	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
44	43	ffvelcdmda 7080	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		prodmo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3163	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3885	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3875	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2854	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3578	. . . . . 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 7014	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2869	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6817	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
57	15, 56	mpbird 260	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
58		f1of 6821	. . . . . . . . . 10 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
59	57, 58	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
60		fvco3 6982	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
64	59	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7276	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtrd 2804	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝐾‘𝑖))
68	67	csbeq1d 3865	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
69	68	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
70		f1of 6821	. . . . . . 7 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
71	36, 70	syl 18	. . . . . 6 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
72	71	ffvelcdmda 7080	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
73		elfznn 13580	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
74		fveq2 6882	. . . . . . 7 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑗) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
75	74	csbeq1d 3865	. . . . . 6 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
76	75, 37	fvmpti 6989	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
77	72, 73, 76	3syl 19	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
78		elfznn 13580	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
79	78	adantl 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80		fveq2 6882	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
81	80	csbeq1d 3865	. . . . . 6 ⊢ (𝑗 = 𝑖 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
82		prodmolem3.4	. . . . . 6 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
83	81, 82	fvmpti 6989	. . . . 5 ⊢ (𝑖 ∈ ℕ → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
84	79, 83	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
85	69, 77, 84	3eqtr4rd 2815	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 14078	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
87	33	fveq2d 6886	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2806	1 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ⦋csb 3861 ifcif 4492 class class class wbr 5113 ↦ cmpt 5196 I cid 5556 ^◡ccnv 5661 ∘ ccom 5666 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7411 ≈ cen 8939 Fincfn 8942 ℂcc 11097 1c1 11100 · cmul 11104 ℕcn 12232 ℕ₀cn0 12503 ℤcz 12590 ℤ_≥cuz 12861 ...cfz 13534 seqcseq 14036 ♯chash 14365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-fzo 13682 df-seq 14037 df-hash 14366

This theorem is referenced by: prodmolem2a 15987 prodmo 15989

W3C validator