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 42819 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 1922 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climrecf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
9 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6727 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
11 | 10 | nfel1 2920 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ (ℂ ∖ {0}) |
12 | 7, 11 | nfim 1904 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
13 | eleq1w 2820 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
14 | 13 | anbi2d 632 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
15 | fveq2 6717 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
16 | 15 | eleq1d 2822 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) ↔ (𝐺‘𝑗) ∈ (ℂ ∖ {0}))) |
17 | 14, 16 | imbi12d 348 | . . 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 6727 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
22 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘1 | |
23 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘 / | |
24 | 22, 23, 10 | nfov 7243 | . . . . 5 ⊢ Ⅎ𝑘(1 / (𝐺‘𝑗)) |
25 | 21, 24 | nfeq 2917 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = (1 / (𝐺‘𝑗)) |
26 | 7, 25 | nfim 1904 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
27 | fveq2 6717 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
28 | 15 | oveq2d 7229 | . . . . 5 ⊢ (𝑘 = 𝑗 → (1 / (𝐺‘𝑘)) = (1 / (𝐺‘𝑗))) |
29 | 27, 28 | eqeq12d 2753 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = (1 / (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = (1 / (𝐺‘𝑗)))) |
30 | 14, 29 | imbi12d 348 | . . 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 42819 | 1 ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 ≠ wne 2940 ∖ cdif 3863 {csn 4541 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 / cdiv 11489 ℤcz 12176 ℤ≥cuz 12438 ⇝ cli 15045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 |
This theorem is referenced by: climdivf 42828 |
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