Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climrecf | Structured version Visualization version GIF version |
Description: A version of climrec 41877 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 1911 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climrecf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
9 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6674 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
11 | 10 | nfel1 2994 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ (ℂ ∖ {0}) |
12 | 7, 11 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
13 | eleq1w 2895 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
14 | 13 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
15 | fveq2 6664 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
16 | 15 | eleq1d 2897 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) ↔ (𝐺‘𝑗) ∈ (ℂ ∖ {0}))) |
17 | 14, 16 | imbi12d 347 | . . 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 | chvarfv 2238 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
20 | climrecf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
21 | 20, 9 | nffv 6674 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
22 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘1 | |
23 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘 / | |
24 | 22, 23, 10 | nfov 7180 | . . . . 5 ⊢ Ⅎ𝑘(1 / (𝐺‘𝑗)) |
25 | 21, 24 | nfeq 2991 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = (1 / (𝐺‘𝑗)) |
26 | 7, 25 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
27 | fveq2 6664 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
28 | 15 | oveq2d 7166 | . . . . 5 ⊢ (𝑘 = 𝑗 → (1 / (𝐺‘𝑘)) = (1 / (𝐺‘𝑗))) |
29 | 27, 28 | eqeq12d 2837 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = (1 / (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = (1 / (𝐺‘𝑗)))) |
30 | 14, 29 | imbi12d 347 | . . 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 | chvarfv 2238 | . 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 41877 | 1 ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ≠ wne 3016 ∖ cdif 3932 {csn 4560 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 / cdiv 11291 ℤcz 11975 ℤ≥cuz 12237 ⇝ cli 14835 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 |
This theorem is referenced by: climdivf 41886 |
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