Theorem prodmolem3 15889

Description: Lemma for prodmo 15892. (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 11113	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
2	1	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3		mulcom 11115	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
4	3	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5		mulass 11117	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
6	5	adantl 482	. . 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 495	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 12818	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10	8, 9	eleqtrdi 2849	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
11		ssidd 3938	. . 3 ⊢ (𝜑 → ℂ ⊆ ℂ)
12		prodmolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
13		f1ocnv 6779	. . . . . 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 6790	. . . . 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 590	. . . 4 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18		ovex 7389	. . . . . . . . . 10 ⊢ (1...𝑁) ∈ V
19	18	f1oen 8909	. . . . . . . . 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 13925	. . . . . . . . 9 ⊢ (1...𝑁) ∈ Fin
22		fzfi 13925	. . . . . . . . 9 ⊢ (1...𝑀) ∈ Fin
23		hashen 14300	. . . . . . . . 9 ⊢ (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
24	21, 22, 23	mp2an 698	. . . . . . . 8 ⊢ ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
25	20, 24	sylibr 235	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
26	7	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
27	26	nnnn0d 12489	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
28		hashfz1 14299	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
29	27, 28	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30	8	nnnn0d 12489	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
31		hashfz1 14299	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
33	25, 29, 32	3eqtr3rd 2783	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
34	33	oveq2d 7372	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
35	34	f1oeq2d 6763	. . . 4 ⊢ (𝜑 → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
36	17, 35	mpbird 258	. . 3 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37		prodmo.3	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
38		fveq2 6827	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑓‘𝑗) = (𝑓‘𝑚))
39	38	csbeq1d 3835	. . . . 5 ⊢ (𝑗 = 𝑚 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
40		elfznn 13498	. . . . . 6 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
41	40	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42		f1of 6767	. . . . . . . 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 7025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
45		prodmo.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
46	45	ralrimiva 3131	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
47	46	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
48		nfcsb1v 3855	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
49	48	nfel1 2917	. . . . . . 7 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
50		csbeq1a 3845	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
51	50	eleq1d 2824	. . . . . . 7 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
52	49, 51	rspc 3548	. . . . . 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 6959	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
55	54, 53	eqeltrd 2839	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
56	34	f1oeq2d 6763	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
57	15, 56	mpbird 258	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
58		f1of 6767	. . . . . . . . . 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 6927	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
61	59, 60	sylan 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
62	61	fveq2d 6831	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(◡𝑓‘(𝐾‘𝑖))))
63	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
64	59	ffvelcdmda 7025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
65		f1ocnvfv2 7221	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
66	63, 64, 65	syl2anc 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
67	62, 66	eqtrd 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝐾‘𝑖))
68	67	csbeq1d 3835	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
69	68	fveq2d 6831	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ( I ‘⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵) = ( I ‘⦋(𝐾‘𝑖) / 𝑘⦌𝐵))
70		f1of 6767	. . . . . . 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 7025	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
73		elfznn 13498	. . . . 5 ⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
74		fveq2 6827	. . . . . . 7 ⊢ (𝑗 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑗) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
75	74	csbeq1d 3835	. . . . . 6 ⊢ (𝑗 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
76	75, 37	fvmpti 6934	. . . . 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 13498	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
79	78	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80		fveq2 6827	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
81	80	csbeq1d 3835	. . . . . 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 6934	. . . . 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 2785	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)))
86	2, 4, 6, 10, 11, 36, 55, 85	seqf1o 13996	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
87	33	fveq2d 6831	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
88	86, 87	eqtr3d 2776	1 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ⦋csb 3831  ifcif 4454   class class class wbr 5072   ↦ cmpt 5153   I cid 5512  ^◡ccnv 5617   ∘ ccom 5622  ⟶wf 6481  –1-1-onto→wf1o 6484  ‘cfv 6485  (class class class)co 7356   ≈ cen 8880  Fincfn 8883  ℂcc 11027  1c1 11030   · cmul 11034  ℕcn 12165  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  ...cfz 13452  seqcseq 13954  ♯chash 14283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284

This theorem is referenced by:  prodmolem2a  15890  prodmo  15892