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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 15664 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 1911 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
8 | 6, 7 | nfan 1896 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
9 | climaddf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
10 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
11 | 9, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
12 | 11 | nfel1 2919 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
13 | 8, 12 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
14 | eleq1w 2821 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍)) | |
15 | 14 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍))) |
16 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
17 | 16 | eleq1d 2823 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ)) |
18 | 15, 17 | imbi12d 344 | . . 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 | chvarfv 2237 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
21 | climaddf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
22 | 21, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
23 | 22 | nfel1 2919 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
24 | 8, 23 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
25 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
26 | 25 | eleq1d 2823 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ)) |
27 | 15, 26 | imbi12d 344 | . . 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 | chvarfv 2237 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
30 | climaddf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
31 | 30, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
32 | nfcv 2902 | . . . . . 6 ⊢ Ⅎ𝑘 + | |
33 | 11, 32, 22 | nfov 7460 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) + (𝐺‘𝑗)) |
34 | 31, 33 | nfeq 2916 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)) |
35 | 8, 34 | nfim 1893 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
36 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗)) | |
37 | 16, 25 | oveq12d 7448 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) + (𝐺‘𝑘)) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
38 | 36, 37 | eqeq12d 2750 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)))) |
39 | 15, 38 | imbi12d 344 | . . 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 | chvarfv 2237 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
42 | 1, 2, 3, 4, 5, 20, 29, 41 | climadd 15664 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 + caddc 11155 ℤcz 12610 ℤ≥cuz 12875 ⇝ cli 15516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 |
This theorem is referenced by: fourierdlem112 46173 |
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