![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > climmulf | Structured version Visualization version GIF version |
Description: A version of climmul 14981 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 2955 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 7 | nfel1 2971 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 |
9 | 6, 8 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | climmulf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | 10, 7 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
12 | 11 | nfel1 2971 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
13 | 9, 12 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
14 | eleq1w 2872 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
15 | 14 | anbi2d 631 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
16 | fveq2 6645 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
17 | 16 | eleq1d 2874 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
18 | 15, 17 | imbi12d 348 | . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
21 | climmulf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
22 | 21, 7 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
23 | 22 | nfel1 2971 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
24 | 9, 23 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
25 | fveq2 6645 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
26 | 25 | eleq1d 2874 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ)) |
27 | 15, 26 | imbi12d 348 | . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
30 | climmulf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
31 | 30, 7 | nffv 6655 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
32 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑘 · | |
33 | 11, 32, 22 | nfov 7165 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) · (𝐺‘𝑗)) |
34 | 31, 33 | nfeq 2968 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)) |
35 | 9, 34 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
36 | fveq2 6645 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
37 | 16, 25 | oveq12d 7153 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) · (𝐺‘𝑘)) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
38 | 36, 37 | eqeq12d 2814 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))) |
39 | 15, 38 | imbi12d 348 | . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
42 | 1, 2, 3, 4, 5, 20, 29, 41 | climmul 14981 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 · cmul 10531 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14833 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 |
This theorem is referenced by: climneg 42252 climdivf 42254 stirlinglem15 42730 etransclem48 42924 |
Copyright terms: Public domain | W3C validator |