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Mirrors > Home > MPE Home > Th. List > Mathboxes > climmulf | Structured version Visualization version GIF version |
Description: A version of climmul 14977 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
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))) |
Ref | Expression |
---|---|
climmulf | ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) |
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 2974 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 7 | nfel1 2991 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 |
9 | 6, 8 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | climmulf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | 10, 7 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
12 | 11 | nfel1 2991 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
13 | 9, 12 | nfim 1888 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
14 | eleq1w 2892 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
15 | 14 | anbi2d 628 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
16 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
17 | 16 | eleq1d 2894 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
18 | 15, 17 | imbi12d 346 | . . 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 2232 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
21 | climmulf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
22 | 21, 7 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
23 | 22 | nfel1 2991 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
24 | 9, 23 | nfim 1888 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
25 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
26 | 25 | eleq1d 2894 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ)) |
27 | 15, 26 | imbi12d 346 | . . 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 2232 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
30 | climmulf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
31 | 30, 7 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
32 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑘 · | |
33 | 11, 32, 22 | nfov 7175 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) · (𝐺‘𝑗)) |
34 | 31, 33 | nfeq 2988 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)) |
35 | 9, 34 | nfim 1888 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
36 | fveq2 6663 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
37 | 16, 25 | oveq12d 7163 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) · (𝐺‘𝑘)) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
38 | 36, 37 | eqeq12d 2834 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))) |
39 | 15, 38 | imbi12d 346 | . . 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 2232 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
42 | 1, 2, 3, 4, 5, 20, 29, 41 | climmul 14977 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 · cmul 10530 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 |
This theorem is referenced by: climneg 41767 climdivf 41769 stirlinglem15 42250 etransclem48 42444 |
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