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Theorem prodmolem3 15963

Description: Lemma for prodmo 15966. (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 11157	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
2	1	adantl 485	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3		mulcom 11159	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
4	3	adantl 485	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5		mulass 11161	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
6	5	adantl 485	. . 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 498	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 12878	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2872	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		ssidd 3959	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		prodmolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
13		f1ocnv 6819	. . . . . 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 6830	. . . . 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 593	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18		ovex 7429	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 8953	. . . . . . . . 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 13985	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 13985	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14360	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 702	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 236	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12542	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
28		hashfz1 14359	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30	8	nnnn0d 12542	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
31		hashfz1 14359	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2806	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7412	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6802	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
36	17, 35	mpbird 259	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37		prodmo.3	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
38		fveq2 6867	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑓‘𝑗) = (𝑓‘𝑚))
39	38	csbeq1d 3856	. . . . 5 ⊢ (𝑗 = 𝑚 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13558	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6806	. . . . . . . 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 7065	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		prodmo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3154	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3876	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2940	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3866	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2847	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3569	. . . . . 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 6999	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2862	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6802	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
57	15, 56	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
58		f1of 6806	. . . . . . . . . 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 6967	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6871	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
64	59	ffvelcdmda 7065	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7261	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtrd 2797	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝐾‘𝑖))
68	67	csbeq1d 3856	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
69	68	fveq2d 6871	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
70		f1of 6806	. . . . . . 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 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
73		elfznn 13558	. . . . 5 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
74		fveq2 6867	. . . . . . 7 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑗) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
75	74	csbeq1d 3856	. . . . . 6 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
76	75, 37	fvmpti 6974	. . . . 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 13558	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
79	78	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80		fveq2 6867	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
81	80	csbeq1d 3856	. . . . . 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 6974	. . . . 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 2808	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 14056	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
87	33	fveq2d 6871	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2799	1 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ⦋csb 3852 ifcif 4480 class class class wbr 5100 ↦ cmpt 5181 I cid 5541 ^◡ccnv 5646 ∘ ccom 5651 ⟶wf 6517 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 ≈ cen 8924 Fincfn 8927 ℂcc 11071 1c1 11074 · cmul 11078 ℕcn 12210 ℕ₀cn0 12481 ℤcz 12568 ℤ_≥cuz 12839 ...cfz 13512 seqcseq 14014 ♯chash 14343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-nn 12211 df-n0 12482 df-z 12569 df-uz 12840 df-fz 13513 df-fzo 13660 df-seq 14015 df-hash 14344

This theorem is referenced by: prodmolem2a 15964 prodmo 15966

W3C validator