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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climmulf | Structured version Visualization version GIF version |
Description: A version of climmul 14701 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 2939 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
8 | 7 | nfel1 2954 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 |
9 | 6, 8 | nfan 1999 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
10 | climmulf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
11 | 10, 7 | nffv 6419 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
12 | 11 | nfel1 2954 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
13 | 9, 12 | nfim 1996 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
14 | eleq1w 2859 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
15 | 14 | anbi2d 623 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
16 | fveq2 6409 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
17 | 16 | eleq1d 2861 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
18 | 15, 17 | imbi12d 336 | . . 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 | chvar 2379 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
21 | climmulf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
22 | 21, 7 | nffv 6419 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
23 | 22 | nfel1 2954 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
24 | 9, 23 | nfim 1996 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
25 | fveq2 6409 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
26 | 25 | eleq1d 2861 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ)) |
27 | 15, 26 | imbi12d 336 | . . 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 | chvar 2379 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
30 | climmulf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
31 | 30, 7 | nffv 6419 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
32 | nfcv 2939 | . . . . . 6 ⊢ Ⅎ𝑘 · | |
33 | 11, 32, 22 | nfov 6906 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) · (𝐺‘𝑗)) |
34 | 31, 33 | nfeq 2951 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)) |
35 | 9, 34 | nfim 1996 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
36 | fveq2 6409 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
37 | 16, 25 | oveq12d 6894 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) · (𝐺‘𝑘)) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
38 | 36, 37 | eqeq12d 2812 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) · (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗)))) |
39 | 15, 38 | imbi12d 336 | . . 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 | chvar 2379 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) · (𝐺‘𝑗))) |
42 | 1, 2, 3, 4, 5, 20, 29, 41 | climmul 14701 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 · 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 Ⅎwnf 1879 ∈ wcel 2157 Ⅎwnfc 2926 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 ℂcc 10220 · cmul 10227 ℤcz 11662 ℤ≥cuz 11926 ⇝ cli 14553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-n0 11577 df-z 11663 df-uz 11927 df-rp 12071 df-seq 13052 df-exp 13111 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-clim 14557 |
This theorem is referenced by: climneg 40574 climdivf 40576 stirlinglem15 41036 etransclem48 41230 |
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