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Theorem fprodser 15889

Description: A finite product expressed in terms of a partial product of an infinite sequence. The recursive definition of a finite product follows from here. (Contributed by Scott Fenton, 14-Dec-2017.)

**Hypotheses**
Ref	Expression
fprodser.1	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴)
fprodser.2	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
fprodser.3	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)

**Assertion**
Ref	Expression
fprodser	⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( · , 𝐹)‘𝑁))

Distinct variable groups: 𝑘,𝐹 𝜑,𝑘 𝑘,𝑀 𝑘,𝑁 Allowed substitution hint: 𝐴(𝑘)

Proof of Theorem fprodser Dummy variables 𝑗 𝑚 𝑛 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfc 15885	. 2 ⊢ ∏𝑗 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑗) = ∏𝑘 ∈ (𝑀...𝑁)𝐴
2		fveq2 6888	. . . 4 ⊢ (𝑗 = ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑗) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)))
3		fprodser.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		eluzelz 12828	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
5	3, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ)
6	5	zcnd 12663	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
7		eluzel2 12823	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
8	3, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	8	zcnd 12663	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ)
10		1cnd 11205	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
11	6, 9, 10	subadd23d 11589	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (𝑁 + (1 − 𝑀)))
12	11	eqcomd 2738	. . . . 5 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) = ((𝑁 − 𝑀) + 1))
13		uznn0sub 12857	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
14	3, 13	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
15		nn0p1nn 12507	. . . . . 6 ⊢ ((𝑁 − 𝑀) ∈ ℕ₀ → ((𝑁 − 𝑀) + 1) ∈ ℕ)
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑁 − 𝑀) + 1) ∈ ℕ)
17	12, 16	eqeltrd 2833	. . . 4 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℕ)
18	10, 9	pncan3d 11570	. . . . . . . . . . 11 ⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀)
19	6, 10, 9	pnpncand 11631	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 + (1 − 𝑀)) + (𝑀 − 1)) = 𝑁)
20	18, 19	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1))) = (𝑀...𝑁))
21	20	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜑 → (𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1))) ↔ 𝑝 ∈ (𝑀...𝑁)))
22	21	biimpa 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))) → 𝑝 ∈ (𝑀...𝑁))
23		elfzelz 13497	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (𝑀...𝑁) → 𝑝 ∈ ℤ)
24	23	zcnd 12663	. . . . . . . . . . . 12 ⊢ (𝑝 ∈ (𝑀...𝑁) → 𝑝 ∈ ℂ)
25	24	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑝 ∈ ℂ)
26		peano2zm 12601	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈ ℤ)
27	8, 26	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 − 1) ∈ ℤ)
28	27	zcnd 12663	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 − 1) ∈ ℂ)
29	28	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → (𝑀 − 1) ∈ ℂ)
30	25, 29	npcand 11571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)) = 𝑝)
31		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑝 ∈ (𝑀...𝑁))
32	30, 31	eqeltrd 2833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)) ∈ (𝑀...𝑁))
33		ovex 7438	. . . . . . . . . 10 ⊢ (𝑝 − (𝑀 − 1)) ∈ V
34		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑝 − (𝑀 − 1)) → (𝑛 + (𝑀 − 1)) = ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)))
35	34	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑛 = (𝑝 − (𝑀 − 1)) → ((𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁) ↔ ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)) ∈ (𝑀...𝑁)))
36	33, 35	sbcie 3819	. . . . . . . . 9 ⊢ ([(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁) ↔ ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)) ∈ (𝑀...𝑁))
37	32, 36	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → [(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁))
38	22, 37	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))) → [(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁))
39	38	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))[(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁))
40		1zzd 12589	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ)
41	17	nnzd 12581	. . . . . . 7 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ ℤ)
42		fzshftral 13585	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ (𝑁 + (1 − 𝑀)) ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → (∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))(𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁) ↔ ∀𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))[(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁)))
43	40, 41, 27, 42	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))(𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁) ↔ ∀𝑝 ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))[(𝑝 − (𝑀 − 1)) / 𝑛](𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁)))
44	39, 43	mpbird 256	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))(𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁))
45	8	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ)
46	5	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ)
47	23	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑝 ∈ ℤ)
48	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → (𝑀 − 1) ∈ ℤ)
49		fzsubel 13533	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑝 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ)) → (𝑝 ∈ (𝑀...𝑁) ↔ (𝑝 − (𝑀 − 1)) ∈ ((𝑀 − (𝑀 − 1))...(𝑁 − (𝑀 − 1)))))
50	45, 46, 47, 48, 49	syl22anc 837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → (𝑝 ∈ (𝑀...𝑁) ↔ (𝑝 − (𝑀 − 1)) ∈ ((𝑀 − (𝑀 − 1))...(𝑁 − (𝑀 − 1)))))
51	31, 50	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → (𝑝 − (𝑀 − 1)) ∈ ((𝑀 − (𝑀 − 1))...(𝑁 − (𝑀 − 1))))
52	9, 10	nncand 11572	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 − (𝑀 − 1)) = 1)
53	6, 9, 10	subsub2d 11596	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − (𝑀 − 1)) = (𝑁 + (1 − 𝑀)))
54	52, 53	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 − (𝑀 − 1))...(𝑁 − (𝑀 − 1))) = (1...(𝑁 + (1 − 𝑀))))
55	54	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ((𝑀 − (𝑀 − 1))...(𝑁 − (𝑀 − 1))) = (1...(𝑁 + (1 − 𝑀))))
56	51, 55	eleqtrd 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → (𝑝 − (𝑀 − 1)) ∈ (1...(𝑁 + (1 − 𝑀))))
57	30	eqcomd 2738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → 𝑝 = ((𝑝 − (𝑀 − 1)) + (𝑀 − 1)))
58	34	rspceeqv 3632	. . . . . . . 8 ⊢ (((𝑝 − (𝑀 − 1)) ∈ (1...(𝑁 + (1 − 𝑀))) ∧ 𝑝 = ((𝑝 − (𝑀 − 1)) + (𝑀 − 1))) → ∃𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)))
59	56, 57, 58	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ∃𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)))
60		elfzelz 13497	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) → 𝑛 ∈ ℤ)
61	60	zcnd 12663	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) → 𝑛 ∈ ℂ)
62		elfzelz 13497	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...(𝑁 + (1 − 𝑀))) → 𝑚 ∈ ℤ)
63	62	zcnd 12663	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(𝑁 + (1 − 𝑀))) → 𝑚 ∈ ℂ)
64	61, 63	anim12i 613	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ))
65		eqtr2 2756	. . . . . . . . . . 11 ⊢ ((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → (𝑛 + (𝑀 − 1)) = (𝑚 + (𝑀 − 1)))
66		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → 𝑛 ∈ ℂ)
67		simprr 771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → 𝑚 ∈ ℂ)
68	28	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → (𝑀 − 1) ∈ ℂ)
69	66, 67, 68	addcan2d 11414	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → ((𝑛 + (𝑀 − 1)) = (𝑚 + (𝑀 − 1)) ↔ 𝑛 = 𝑚))
70	65, 69	imbitrid 243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ℂ ∧ 𝑚 ∈ ℂ)) → ((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → 𝑛 = 𝑚))
71	64, 70	sylan2 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀))))) → ((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → 𝑛 = 𝑚))
72	71	ralrimivva 3200	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))∀𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → 𝑛 = 𝑚))
73	72	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))∀𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → 𝑛 = 𝑚))
74		oveq1 7412	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 + (𝑀 − 1)) = (𝑚 + (𝑀 − 1)))
75	74	eqeq2d 2743	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑝 = (𝑛 + (𝑀 − 1)) ↔ 𝑝 = (𝑚 + (𝑀 − 1))))
76	75	reu4 3726	. . . . . . 7 ⊢ (∃!𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)) ↔ (∃𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))∀𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))((𝑝 = (𝑛 + (𝑀 − 1)) ∧ 𝑝 = (𝑚 + (𝑀 − 1))) → 𝑛 = 𝑚)))
77	59, 73, 76	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑝 ∈ (𝑀...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)))
78	77	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑝 ∈ (𝑀...𝑁)∃!𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1)))
79		eqid 2732	. . . . . 6 ⊢ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))) = (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))
80	79	f1ompt 7107	. . . . 5 ⊢ ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁) ↔ (∀𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))(𝑛 + (𝑀 − 1)) ∈ (𝑀...𝑁) ∧ ∀𝑝 ∈ (𝑀...𝑁)∃!𝑛 ∈ (1...(𝑁 + (1 − 𝑀)))𝑝 = (𝑛 + (𝑀 − 1))))
81	44, 78, 80	sylanbrc 583	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁))
82		fprodser.3	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ)
83	82	fmpttd 7111	. . . . 5 ⊢ (𝜑 → (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴):(𝑀...𝑁)⟶ℂ)
84	83	ffvelcdmda 7083	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑗) ∈ ℂ)
85		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → 𝑚 ∈ (1...(𝑁 + (1 − 𝑀))))
86		1zzd 12589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → 1 ∈ ℤ)
87	41	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑁 + (1 − 𝑀)) ∈ ℤ)
88	62	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → 𝑚 ∈ ℤ)
89	27	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑀 − 1) ∈ ℤ)
90		fzaddel 13531	. . . . . . . . 9 ⊢ (((1 ∈ ℤ ∧ (𝑁 + (1 − 𝑀)) ∈ ℤ) ∧ (𝑚 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ)) → (𝑚 ∈ (1...(𝑁 + (1 − 𝑀))) ↔ (𝑚 + (𝑀 − 1)) ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))))
91	86, 87, 88, 89, 90	syl22anc 837	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑚 ∈ (1...(𝑁 + (1 − 𝑀))) ↔ (𝑚 + (𝑀 − 1)) ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1)))))
92	85, 91	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑚 + (𝑀 − 1)) ∈ ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1))))
93	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((1 + (𝑀 − 1))...((𝑁 + (1 − 𝑀)) + (𝑀 − 1))) = (𝑀...𝑁))
94	92, 93	eleqtrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁))
95		fprodser.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴)
96	95	ralrimiva 3146	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = 𝐴)
97		nfcsb1v 3917	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴
98	97	nfeq2 2920	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐹‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴
99		fveq2 6888	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + (𝑀 − 1)) → (𝐹‘𝑘) = (𝐹‘(𝑚 + (𝑀 − 1))))
100		csbeq1a 3906	. . . . . . . . 9 ⊢ (𝑘 = (𝑚 + (𝑀 − 1)) → 𝐴 = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
101	99, 100	eqeq12d 2748	. . . . . . . 8 ⊢ (𝑘 = (𝑚 + (𝑀 − 1)) → ((𝐹‘𝑘) = 𝐴 ↔ (𝐹‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴))
102	98, 101	rspc 3600	. . . . . . 7 ⊢ ((𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)(𝐹‘𝑘) = 𝐴 → (𝐹‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴))
103	96, 102	mpan9 507	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁)) → (𝐹‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
104	94, 103	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝐹‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
105		f1of 6830	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))–1-1-onto→(𝑀...𝑁) → (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
106	81, 105	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁))
107		fvco3 6987	. . . . . . 7 ⊢ (((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))):(1...(𝑁 + (1 − 𝑀)))⟶(𝑀...𝑁) ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))))‘𝑚) = (𝐹‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)))
108	106, 107	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))))‘𝑚) = (𝐹‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)))
109		ovex 7438	. . . . . . . . 9 ⊢ (𝑚 + (𝑀 − 1)) ∈ V
110	74, 79, 109	fvmpt 6995	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(𝑁 + (1 − 𝑀))) → ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚) = (𝑚 + (𝑀 − 1)))
111	110	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚) = (𝑚 + (𝑀 − 1)))
112	111	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → (𝐹‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)) = (𝐹‘(𝑚 + (𝑀 − 1))))
113	108, 112	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))))‘𝑚) = (𝐹‘(𝑚 + (𝑀 − 1))))
114	111	fveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘(𝑚 + (𝑀 − 1))))
115	82	ralrimiva 3146	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ)
116	97	nfel1 2919	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ
117	100	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑘 = (𝑚 + (𝑀 − 1)) → (𝐴 ∈ ℂ ↔ ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ))
118	116, 117	rspc 3600	. . . . . . . . 9 ⊢ ((𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁) → (∀𝑘 ∈ (𝑀...𝑁)𝐴 ∈ ℂ → ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ))
119	115, 118	mpan9 507	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁)) → ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ)
120	94, 119	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ)
121		eqid 2732	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴) = (𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)
122	121	fvmpts 6998	. . . . . . 7 ⊢ (((𝑚 + (𝑀 − 1)) ∈ (𝑀...𝑁) ∧ ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴 ∈ ℂ) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
123	94, 120, 122	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘(𝑚 + (𝑀 − 1))) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
124	114, 123	eqtrd 2772	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)) = ⦋(𝑚 + (𝑀 − 1)) / 𝑘⦌𝐴)
125	104, 113, 124	3eqtr4d 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...(𝑁 + (1 − 𝑀)))) → ((𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1))))‘𝑚) = ((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘((𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))‘𝑚)))
126	2, 17, 81, 84, 125	fprod 15881	. . 3 ⊢ (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑗) = (seq1( · , (𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))))‘(𝑁 + (1 − 𝑀))))
127		nnuz 12861	. . . . 5 ⊢ ℕ = (ℤ_≥‘1)
128	17, 127	eleqtrdi 2843	. . . 4 ⊢ (𝜑 → (𝑁 + (1 − 𝑀)) ∈ (ℤ_≥‘1))
129	128, 27, 113	seqshft2 13990	. . 3 ⊢ (𝜑 → (seq1( · , (𝐹 ∘ (𝑛 ∈ (1...(𝑁 + (1 − 𝑀))) ↦ (𝑛 + (𝑀 − 1)))))‘(𝑁 + (1 − 𝑀))) = (seq(1 + (𝑀 − 1))( · , 𝐹)‘((𝑁 + (1 − 𝑀)) + (𝑀 − 1))))
130	18	seqeq1d 13968	. . . 4 ⊢ (𝜑 → seq(1 + (𝑀 − 1))( · , 𝐹) = seq𝑀( · , 𝐹))
131	130, 19	fveq12d 6895	. . 3 ⊢ (𝜑 → (seq(1 + (𝑀 − 1))( · , 𝐹)‘((𝑁 + (1 − 𝑀)) + (𝑀 − 1))) = (seq𝑀( · , 𝐹)‘𝑁))
132	126, 129, 131	3eqtrd 2776	. 2 ⊢ (𝜑 → ∏𝑗 ∈ (𝑀...𝑁)((𝑘 ∈ (𝑀...𝑁) ↦ 𝐴)‘𝑗) = (seq𝑀( · , 𝐹)‘𝑁))
133	1, 132	eqtr3id 2786	1 ⊢ (𝜑 → ∏𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( · , 𝐹)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ∃!wreu 3374 [wsbc 3776 ⦋csb 3892 ↦ cmpt 5230 ∘ ccom 5679 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11440 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℤ_≥cuz 12818 ...cfz 13480 seqcseq 13962 ∏cprod 15845

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-prod 15846

This theorem is referenced by: fprodfac 15913 iprodclim3 15940

W3C validator