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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 45129 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 1909 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climrecf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
9 | nfcv 2891 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6906 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
11 | 10 | nfel1 2908 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ (ℂ ∖ {0}) |
12 | 7, 11 | nfim 1891 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
13 | eleq1w 2808 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
14 | 13 | anbi2d 628 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
15 | fveq2 6896 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
16 | 15 | eleq1d 2810 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) ↔ (𝐺‘𝑗) ∈ (ℂ ∖ {0}))) |
17 | 14, 16 | imbi12d 343 | . . 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 2228 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
20 | climrecf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
21 | 20, 9 | nffv 6906 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
22 | nfcv 2891 | . . . . . 6 ⊢ Ⅎ𝑘1 | |
23 | nfcv 2891 | . . . . . 6 ⊢ Ⅎ𝑘 / | |
24 | 22, 23, 10 | nfov 7449 | . . . . 5 ⊢ Ⅎ𝑘(1 / (𝐺‘𝑗)) |
25 | 21, 24 | nfeq 2905 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = (1 / (𝐺‘𝑗)) |
26 | 7, 25 | nfim 1891 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
27 | fveq2 6896 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
28 | 15 | oveq2d 7435 | . . . . 5 ⊢ (𝑘 = 𝑗 → (1 / (𝐺‘𝑘)) = (1 / (𝐺‘𝑗))) |
29 | 27, 28 | eqeq12d 2741 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = (1 / (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = (1 / (𝐺‘𝑗)))) |
30 | 14, 29 | imbi12d 343 | . . 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 2228 | . 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 45129 | 1 ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ≠ wne 2929 ∖ cdif 3941 {csn 4630 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 / cdiv 11903 ℤcz 12591 ℤ≥cuz 12855 ⇝ cli 15464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-rp 13010 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 |
This theorem is referenced by: climdivf 45138 |
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