Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > climaddf | Structured version Visualization version GIF version |
Description: A version of climadd 14362 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
climaddf.1 | ⊢ Ⅎ𝑘𝜑 |
climaddf.2 | ⊢ Ⅎ𝑘𝐹 |
climaddf.3 | ⊢ Ⅎ𝑘𝐺 |
climaddf.4 | ⊢ Ⅎ𝑘𝐻 |
climaddf.5 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
climaddf.6 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
climaddf.7 | ⊢ (𝜑 → 𝐹 ⇝ 𝐴) |
climaddf.8 | ⊢ (𝜑 → 𝐻 ∈ 𝑋) |
climaddf.9 | ⊢ (𝜑 → 𝐺 ⇝ 𝐵) |
climaddf.10 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
climaddf.11 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) |
climaddf.12 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) |
Ref | Expression |
---|---|
climaddf | ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | climaddf.5 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | climaddf.6 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | climaddf.7 | . 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴) | |
4 | climaddf.8 | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋) | |
5 | climaddf.9 | . 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐵) | |
6 | climaddf.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
7 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
8 | 6, 7 | nfan 1828 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
9 | climaddf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
10 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
11 | 9, 10 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
12 | 11 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
13 | 8, 12 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
14 | eleq1 2689 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
15 | 14 | anbi2d 740 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
16 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
17 | 16 | eleq1d 2686 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
18 | 15, 17 | imbi12d 334 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ))) |
19 | climaddf.10 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) | |
20 | 13, 18, 19 | chvar 2262 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
21 | climaddf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
22 | 21, 10 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
23 | 22 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
24 | 8, 23 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
25 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
26 | 25 | eleq1d 2686 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ)) |
27 | 15, 26 | imbi12d 334 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ))) |
28 | climaddf.11 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) | |
29 | 24, 27, 28 | chvar 2262 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
30 | climaddf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
31 | 30, 10 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
32 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑘 + | |
33 | 11, 32, 22 | nfov 6676 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) + (𝐺‘𝑗)) |
34 | 31, 33 | nfeq 2776 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)) |
35 | 8, 34 | nfim 1825 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
36 | fveq2 6191 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
37 | 16, 25 | oveq12d 6668 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) + (𝐺‘𝑘)) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
38 | 36, 37 | eqeq12d 2637 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)))) |
39 | 15, 38 | imbi12d 334 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))))) |
40 | climaddf.12 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) | |
41 | 35, 39, 40 | chvar 2262 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
42 | 1, 2, 3, 4, 5, 20, 29, 41 | climadd 14362 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 + caddc 9939 ℤcz 11377 ℤ≥cuz 11687 ⇝ cli 14215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 |
This theorem is referenced by: fourierdlem112 40435 |
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