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Theorem prodmolem3 15907

Description: Lemma for prodmo 15910. (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 𝑗,𝐺 𝑗,𝑘,𝜑 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 𝑖 𝑚 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 11220	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
2	1	adantl 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3		mulcom 11222	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
4	3	adantl 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5		mulass 11224	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
6	5	adantl 480	. . 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 493	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 12893	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2835	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		ssidd 3996	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		prodmolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
13		f1ocnv 6845	. . . . . 6 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
15		prodmolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
16		f1oco 6856	. . . . 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 582	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18		ovex 7448	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 8990	. . . . . . . . 9 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
20	17, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1...𝑁) ≈ (1...𝑀))
21		fzfi 13967	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 13967	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14336	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 690	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 233	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12560	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		hashfz1 14335	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30	8	nnnn0d 12560	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
31		hashfz1 14335	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2774	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7431	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6829	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
36	17, 35	mpbird 256	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37		prodmo.3	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
38		fveq2 6891	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑓‘𝑗) = (𝑓‘𝑚))
39	38	csbeq1d 3889	. . . . 5 ⊢ (𝑗 = 𝑚 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13560	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6833	. . . . . . . 8 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
43	12, 42	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
44	43	ffvelcdmda 7088	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		prodmo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3136	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3910	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2909	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3899	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2810	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3590	. . . . . 6 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
53	44, 47, 52	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
54	37, 39, 41, 53	fvmptd3 7022	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2825	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6829	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
57	15, 56	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
58		f1of 6833	. . . . . . . . . 10 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
59	57, 58	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
60		fvco3 6991	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 578	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
64	59	ffvelcdmda 7088	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7281	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtrd 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝐾‘𝑖))
68	67	csbeq1d 3889	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
69	68	fveq2d 6895	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
70		f1of 6833	. . . . . . 7 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
71	36, 70	syl 17	. . . . . 6 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
72	71	ffvelcdmda 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
73		elfznn 13560	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
74		fveq2 6891	. . . . . . 7 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑗) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
75	74	csbeq1d 3889	. . . . . 6 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
76	75, 37	fvmpti 6998	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
77	72, 73, 76	3syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵))
78		elfznn 13560	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
79	78	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80		fveq2 6891	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
81	80	csbeq1d 3889	. . . . . 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 6998	. . . . 5 ⊢ (𝑖 ∈ ℕ → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
84	79, 83	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
85	69, 77, 84	3eqtr4rd 2776	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 14038	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
87	33	fveq2d 6895	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2767	1 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ⦋csb 3885 ifcif 4524 class class class wbr 5143 ↦ cmpt 5226 I cid 5569 ^◡ccnv 5671 ∘ ccom 5676 ⟶wf 6538 –1-1-onto→wf1o 6541 ‘cfv 6542 (class class class)co 7415 ≈ cen 8957 Fincfn 8960 ℂcc 11134 1c1 11137 · cmul 11141 ℕcn 12240 ℕ₀cn0 12500 ℤcz 12586 ℤ_≥cuz 12850 ...cfz 13514 seqcseq 13996 ♯chash 14319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320

This theorem is referenced by: prodmolem2a 15908 prodmo 15910

W3C validator