Theorem climmulf 41883

Description: A version of climmul 14988 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

Hypotheses

Ref	Expression
climmulf.1	⊢ Ⅎ𝑘𝜑
climmulf.2	⊢ Ⅎ𝑘𝐹
climmulf.3	⊢ Ⅎ𝑘𝐺
climmulf.4	⊢ Ⅎ𝑘𝐻
climmulf.5	⊢ 𝑍 = (ℤ≥‘𝑀)
climmulf.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
climmulf.7	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climmulf.8	⊢ (𝜑 → 𝐻 ∈ 𝑋)
climmulf.9	⊢ (𝜑 → 𝐺 ⇝ 𝐵)
climmulf.10	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
climmulf.11	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
climmulf.12	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))

Assertion

Ref	Expression
climmulf	⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))

Distinct variable group:   𝑘,𝑍

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝐻(𝑘)   𝑀(𝑘)   𝑋(𝑘)

Proof of Theorem climmulf

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climmulf.5	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climmulf.6	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climmulf.7	. 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
4		climmulf.8	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
5		climmulf.9	. 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
6		climmulf.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
7		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘𝑗
8	7	nfel1 2994	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
9	6, 8	nfan 1896	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
10		climmulf.2	. . . . . 6 ⊢ Ⅎ𝑘𝐹
11	10, 7	nffv 6679	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
12	11	nfel1 2994	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ
13	9, 12	nfim 1893	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
14		eleq1w 2895	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
15	14	anbi2d 630	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
16		fveq2 6669	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
17	16	eleq1d 2897	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ))
18	15, 17	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)))
19		climmulf.10	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
20	13, 18, 19	chvarfv 2238	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
21		climmulf.3	. . . . . 6 ⊢ Ⅎ𝑘𝐺
22	21, 7	nffv 6679	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
23	22	nfel1 2994	. . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ
24	9, 23	nfim 1893	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
25		fveq2 6669	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
26	25	eleq1d 2897	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ))
27	15, 26	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)))
28		climmulf.11	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
29	24, 27, 28	chvarfv 2238	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
30		climmulf.4	. . . . . 6 ⊢ Ⅎ𝑘𝐻
31	30, 7	nffv 6679	. . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗)
32		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘 ·
33	11, 32, 22	nfov 7185	. . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) · (𝐺‘𝑗))
34	31, 33	nfeq 2991	. . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))
35	9, 34	nfim 1893	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
36		fveq2 6669	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
37	16, 25	oveq12d 7173	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) · (𝐺‘𝑘)) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
38	36, 37	eqeq12d 2837	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))))
39	15, 38	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘))) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))))
40		climmulf.12	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)))
41	35, 39, 40	chvarfv 2238	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))
42	1, 2, 3, 4, 5, 20, 29, 41	climmul 14988	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533  Ⅎwnf 1780   ∈ wcel 2110  Ⅎwnfc 2961   class class class wbr 5065  ‘cfv 6354  (class class class)co 7155  ℂcc 10534   · cmul 10541  ℤcz 11980  ℤ_≥cuz 12242   ⇝ cli 14840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-sup 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594  df-clim 14844

This theorem is referenced by:  climneg  41889  climdivf  41891  stirlinglem15  42372  etransclem48  42566