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 38670 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 1830 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
7 | 5, 6 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
8 | climrecf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
9 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
10 | 8, 9 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
11 | 10 | nfel1 2765 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ (ℂ ∖ {0}) |
12 | 7, 11 | nfim 1813 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
13 | eleq1 2676 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
14 | 13 | anbi2d 736 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
15 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
16 | 15 | eleq1d 2672 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ (ℂ ∖ {0}) ↔ (𝐺‘𝑗) ∈ (ℂ ∖ {0}))) |
17 | 14, 16 | imbi12d 333 | . . 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 2250 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ (ℂ ∖ {0})) |
20 | climrecf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
21 | 20, 9 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
22 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑘1 | |
23 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑘 / | |
24 | 22, 23, 10 | nfov 6575 | . . . . 5 ⊢ Ⅎ𝑘(1 / (𝐺‘𝑗)) |
25 | 21, 24 | nfeq 2762 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = (1 / (𝐺‘𝑗)) |
26 | 7, 25 | nfim 1813 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = (1 / (𝐺‘𝑗))) |
27 | fveq2 6103 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
28 | 15 | oveq2d 6565 | . . . . 5 ⊢ (𝑘 = 𝑗 → (1 / (𝐺‘𝑘)) = (1 / (𝐺‘𝑗))) |
29 | 27, 28 | eqeq12d 2625 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = (1 / (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = (1 / (𝐺‘𝑗)))) |
30 | 14, 29 | imbi12d 333 | . . 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 2250 | . 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 38670 | 1 ⊢ (𝜑 → 𝐻 ⇝ (1 / 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ≠ wne 2780 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 / cdiv 10563 ℤcz 11254 ℤ≥cuz 11563 ⇝ cli 14063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 |
This theorem is referenced by: climdivf 38679 |
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