![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > climrecf | Structured version Visualization version GIF version |
Description: A version of climrec 40567 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
climrecf.1 | ⊢ Ⅎ𝑘𝜑 |
climrecf.2 | ⊢ Ⅎ𝑘𝐺 |
climrecf.3 | ⊢ Ⅎ𝑘𝐻 |
climrecf.4 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climrecf.5 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climrecf.6 | ⊢ (𝜑 → 𝐺 ⇝ 𝐴) |
climrecf.7 | ⊢ (𝜑 → 𝐴 ≠ 0) |
climrecf.8 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) |
climrecf.9 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) |
climrecf.10 | ⊢ (𝜑 → 𝐻 ∈ 𝑊) |
Ref | Expression |
---|---|
climrecf | ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climrecf.4 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climrecf.5 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climrecf.6 | . 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐴) | |
4 | climrecf.7 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
5 | climrecf.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
6 | nfv 2010 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1999 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climrecf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
9 | nfcv 2939 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6419 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
11 | 10 | nfel1 2954 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ (ℂ ∖ {0}) |
12 | 7, 11 | nfim 1996 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
13 | eleq1w 2859 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
14 | 13 | anbi2d 623 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
15 | fveq2 6409 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
16 | 15 | eleq1d 2861 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) ↔ (𝐺‘𝑗) ∈ (ℂ ∖ {0}))) |
17 | 14, 16 | imbi12d 336 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})))) |
18 | climrecf.8 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ (ℂ ∖ {0})) | |
19 | 12, 17, 18 | chvar 2379 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
20 | climrecf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
21 | 20, 9 | nffv 6419 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
22 | nfcv 2939 | . . . . . 6 ⊢ Ⅎ𝑘1 | |
23 | nfcv 2939 | . . . . . 6 ⊢ Ⅎ𝑘 / | |
24 | 22, 23, 10 | nfov 6906 | . . . . 5 ⊢ Ⅎ𝑘(1 / (𝐺‘𝑗)) |
25 | 21, 24 | nfeq 2951 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = (1 / (𝐺‘𝑗)) |
26 | 7, 25 | nfim 1996 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
27 | fveq2 6409 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
28 | 15 | oveq2d 6892 | . . . . 5 ⊢ (𝑘 = 𝑗 → (1 / (𝐺‘𝑘)) = (1 / (𝐺‘𝑗))) |
29 | 27, 28 | eqeq12d 2812 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = (1 / (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = (1 / (𝐺‘𝑗)))) |
30 | 14, 29 | imbi12d 336 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))))) |
31 | climrecf.9 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (1 / (𝐺‘𝑘))) | |
32 | 26, 30, 31 | chvar 2379 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
33 | climrecf.10 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑊) | |
34 | 1, 2, 3, 4, 19, 32, 33 | climrec 40567 | 1 ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 Ⅎwnf 1879 ∈ wcel 2157 Ⅎwnfc 2926 ≠ wne 2969 ∖ cdif 3764 {csn 4366 class class class wbr 4841 ‘cfv 6099 (class class class)co 6876 ℂcc 10220 0cc0 10222 1c1 10223 / cdiv 10974 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: climdivf 40576 |
Copyright terms: Public domain | W3C validator |