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Theorem prodmodclem3 11538

Description: Lemma for prodmodc 11541. (Contributed by Scott Fenton, 4-Dec-2017.) (Revised by Jim Kingdon, 11-Apr-2024.)

**Hypotheses**
Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmodc.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))
prodmodclem3.4	⊢ 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))
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
prodmodclem3	⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

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

Proof of Theorem prodmodclem3 Dummy variables 𝑖 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcl 7901	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) ∈ ℂ)
2	1	adantl 275	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) ∈ ℂ)
3		mulcom 7903	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
4	3	adantl 275	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → (𝑚 · 𝑦) = (𝑦 · 𝑚))
5		mulass 7905	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
6	5	adantl 275	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → ((𝑚 · 𝑦) · 𝑥) = (𝑚 · (𝑦 · 𝑥)))
7		prodmolem3.5	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 111	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 9522	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10	8, 9	eleqtrdi 2263	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
11		prodmolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
12		f1ocnv 5455	. . . . . 6 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
13	11, 12	syl 14	. . . . 5 ⊢ (𝜑 → ^◡𝑓:𝐴–1-1-onto→(1...𝑀))
14		prodmolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
15		f1oco 5465	. . . . 5 ⊢ ((^◡𝑓:𝐴–1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto→𝐴) → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
16	13, 14, 15	syl2anc 409	. . . 4 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17	7	ancomd 265	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ))
18	17, 14, 11	nnf1o 11339	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
19	18	oveq2d 5869	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
20	19	f1oeq2d 5438	. . . 4 ⊢ (𝜑 → ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (^◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
21	16, 20	mpbird 166	. . 3 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
22		prodmodc.3	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1))
23		breq1 3992	. . . . . 6 ⊢ (𝑗 = 𝑚 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑚 ≤ (♯‘𝐴)))
24		fveq2 5496	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (𝑓‘𝑗) = (𝑓‘𝑚))
25	24	csbeq1d 3056	. . . . . 6 ⊢ (𝑗 = 𝑚 → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
26	23, 25	ifbieq1d 3548	. . . . 5 ⊢ (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 1))
27		elnnuz 9523	. . . . . . 7 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ_≥‘1))
28	27	biimpri 132	. . . . . 6 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℕ)
29	28	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → 𝑚 ∈ ℕ)
30		f1of 5442	. . . . . . . . . 10 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
31	11, 30	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
32	31	ad2antrr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑓:(1...𝑀)⟶𝐴)
33		1zzd 9239	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℤ)
34	8	nnzd 9333	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
35	34	ad2antrr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 ∈ ℤ)
36		eluzelz 9496	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 𝑚 ∈ ℤ)
37	36	ad2antlr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ ℤ)
38	33, 35, 37	3jca 1172	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ))
39		eluzle 9499	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘1) → 1 ≤ 𝑚)
40	39	ad2antlr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 1 ≤ 𝑚)
41		simpr 109	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ≤ (♯‘𝐴))
42	8	nnnn0d 9188	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
43		hashfz1 10717	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
44	42, 43	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
45		1zzd 9239	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℤ)
46	45, 34	fzfigd 10387	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
47	46, 11	fihasheqf1od 10724	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘(1...𝑀)) = (♯‘𝐴))
48	44, 47	eqtr3d 2205	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 = (♯‘𝐴))
49	48	ad2antrr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑀 = (♯‘𝐴))
50	41, 49	breqtrrd 4017	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ≤ 𝑀)
51	40, 50	jca 304	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑀))
52		elfz2 9972	. . . . . . . . 9 ⊢ (𝑚 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤ 𝑚 ∧ 𝑚 ≤ 𝑀)))
53	38, 51, 52	sylanbrc 415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑀))
54	32, 53	ffvelrnd 5632	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝑓‘𝑚) ∈ 𝐴)
55		prodmo.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
56	55	ralrimiva 2543	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
57	56	ad2antrr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
58		nfcsb1v 3082	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
59	58	nfel1 2323	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
60		csbeq1a 3058	. . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
61	60	eleq1d 2239	. . . . . . . 8 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
62	59, 61	rspc 2828	. . . . . . 7 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
63	54, 57, 62	sylc 62	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
64		1cnd 7936	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ ¬ 𝑚 ≤ (♯‘𝐴)) → 1 ∈ ℂ)
65	29	nnzd 9333	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → 𝑚 ∈ ℤ)
66	48, 34	eqeltrrd 2248	. . . . . . . 8 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
67	66	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → (♯‘𝐴) ∈ ℤ)
68		zdcle 9288	. . . . . . 7 ⊢ ((𝑚 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑚 ≤ (♯‘𝐴))
69	65, 67, 68	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → DECID 𝑚 ≤ (♯‘𝐴))
70	63, 64, 69	ifcldadc 3555	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → if(𝑚 ≤ (♯‘𝐴), ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 1) ∈ ℂ)
71	22, 26, 29, 70	fvmptd3 5589	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → (𝐺‘𝑚) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 1))
72	71, 70	eqeltrd 2247	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → (𝐺‘𝑚) ∈ ℂ)
73		prodmodclem3.4	. . . . 5 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1))
74		fveq2 5496	. . . . . . 7 ⊢ (𝑗 = 𝑚 → (𝐾‘𝑗) = (𝐾‘𝑚))
75	74	csbeq1d 3056	. . . . . 6 ⊢ (𝑗 = 𝑚 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
76	23, 75	ifbieq1d 3548	. . . . 5 ⊢ (𝑗 = 𝑚 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1))
77	14	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
78		f1of 5442	. . . . . . . . 9 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
79	77, 78	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝐾:(1...𝑁)⟶𝐴)
80	19	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (1...𝑀) = (1...𝑁))
81	53, 80	eleqtrd 2249	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → 𝑚 ∈ (1...𝑁))
82	79, 81	ffvelrnd 5632	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → (𝐾‘𝑚) ∈ 𝐴)
83		nfcsb1v 3082	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
84	83	nfel1 2323	. . . . . . . 8 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
85		csbeq1a 3058	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
86	85	eleq1d 2239	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
87	84, 86	rspc 2828	. . . . . . 7 ⊢ ((𝐾‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
88	82, 57, 87	sylc 62	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) ∧ 𝑚 ≤ (♯‘𝐴)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
89	88, 64, 69	ifcldadc 3555	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → if(𝑚 ≤ (♯‘𝐴), ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1) ∈ ℂ)
90	73, 76, 29, 89	fvmptd3 5589	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → (𝐻‘𝑚) = if(𝑚 ≤ (♯‘𝐴), ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 1))
91	90, 89	eqeltrd 2247	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
92	19	f1oeq2d 5438	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
93	14, 92	mpbird 166	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
94		f1of 5442	. . . . . . . . 9 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
95	93, 94	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
96		fvco3 5567	. . . . . . . 8 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
97	95, 96	sylan 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) = (^◡𝑓‘(𝐾‘𝑖)))
98	97	fveq2d 5500	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(^◡𝑓‘(𝐾‘𝑖))))
99	11	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
100	95	ffvelrnda 5631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
101		f1ocnvfv2 5757	. . . . . . 7 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
102	99, 100, 101	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(^◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
103	98, 102	eqtrd 2203	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = (𝐾‘𝑖))
104	103	csbeq1d 3056	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
105		breq1 3992	. . . . . . 7 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑗 ≤ (♯‘𝐴) ↔ ((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴)))
106		fveq2 5496	. . . . . . . 8 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑗) = (𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
107	106	csbeq1d 3056	. . . . . . 7 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑗) / 𝑘⦌𝐵 = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
108	105, 107	ifbieq1d 3548	. . . . . 6 ⊢ (𝑗 = ((^◡𝑓 ∘ 𝐾)‘𝑖) → if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 1) = if(((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴), ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 1))
109		f1of 5442	. . . . . . . . 9 ⊢ ((^◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
110	21, 109	syl 14	. . . . . . . 8 ⊢ (𝜑 → (^◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
111	110	ffvelrnda 5631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
112		elfznn 10010	. . . . . . 7 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
113	111, 112	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
114		elfzle2 9984	. . . . . . . . . 10 ⊢ (((^◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
115	111, 114	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
116	48	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑀 = (♯‘𝐴))
117	115, 116	breqtrd 4015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴))
118	117	iftrued 3533	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴), ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 1) = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
119	56	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
120		nfcsb1v 3082	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵
121	120	nfel1 2323	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ
122		csbeq1a 3058	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
123	122	eleq1d 2239	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
124	121, 123	rspc 2828	. . . . . . . . 9 ⊢ ((𝐾‘𝑖) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
125	100, 119, 124	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ)
126	104, 125	eqeltrd 2247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵 ∈ ℂ)
127	118, 126	eqeltrd 2247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴), ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 1) ∈ ℂ)
128	22, 108, 113, 127	fvmptd3 5589	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = if(((^◡𝑓 ∘ 𝐾)‘𝑖) ≤ (♯‘𝐴), ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 1))
129	128, 118	eqtrd 2203	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((^◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
130		breq1 3992	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (𝑗 ≤ (♯‘𝐴) ↔ 𝑖 ≤ (♯‘𝐴)))
131		fveq2 5496	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖))
132	131	csbeq1d 3056	. . . . . . 7 ⊢ (𝑗 = 𝑖 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
133	130, 132	ifbieq1d 3548	. . . . . 6 ⊢ (𝑗 = 𝑖 → if(𝑗 ≤ (♯‘𝐴), ⦋(𝐾‘𝑗) / 𝑘⦌𝐵, 1) = if(𝑖 ≤ (♯‘𝐴), ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 1))
134		elfznn 10010	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
135	134	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
136		elfzle2 9984	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
137	136	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀)
138	137, 116	breqtrd 4015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ (♯‘𝐴))
139	138	iftrued 3533	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
140	139, 125	eqeltrd 2247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ (♯‘𝐴), ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 1) ∈ ℂ)
141	73, 133, 135, 140	fvmptd3 5589	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ (♯‘𝐴), ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 1))
142	141, 139	eqtrd 2203	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
143	104, 129, 142	3eqtr4rd 2214	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((^◡𝑓 ∘ 𝐾)‘𝑖)))
144	2, 4, 6, 10, 21, 72, 91, 143	seq3f1o 10460	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
145	18	fveq2d 5500	. 2 ⊢ (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
146	144, 145	eqtr3d 2205	1 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 DECID wdc 829 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⦋csb 3049 ifcif 3526 class class class wbr 3989 ↦ cmpt 4050 ^◡ccnv 4610 ∘ ccom 4615 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 ℂcc 7772 1c1 7775 · cmul 7779 ≤ cle 7955 ℕcn 8878 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 ...cfz 9965 seqcseq 10401 ♯chash 10709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-ihash 10710

This theorem is referenced by: prodmodclem2a 11539 prodmodc 11541

W3C validator