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| Mirrors > Home > MPE Home > Th. List > Mathboxes > climaddf | Structured version Visualization version GIF version | ||
| Description: A version of climadd 15668 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 1914 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍 | |
| 8 | 6, 7 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍) |
| 9 | climaddf.2 | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
| 10 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
| 11 | 9, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
| 12 | 11 | nfel1 2922 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ |
| 13 | 8, 12 | nfim 1896 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
| 14 | eleq1w 2824 | . . . . 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 2826 | . . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ) |
| 21 | climaddf.3 | . . . . . 6 ⊢ Ⅎ𝑘𝐺 | |
| 22 | 21, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗) |
| 23 | 22 | nfel1 2922 | . . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ |
| 24 | 8, 23 | nfim 1896 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
| 25 | fveq2 6906 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗)) | |
| 26 | 25 | eleq1d 2826 | . . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ) |
| 30 | climaddf.4 | . . . . . 6 ⊢ Ⅎ𝑘𝐻 | |
| 31 | 30, 10 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗) |
| 32 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑘 + | |
| 33 | 11, 32, 22 | nfov 7461 | . . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) + (𝐺‘𝑗)) |
| 34 | 31, 33 | nfeq 2919 | . . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)) |
| 35 | 8, 34 | nfim 1896 | . . 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 7449 | . . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) + (𝐺‘𝑘)) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
| 38 | 36, 37 | eqeq12d 2753 | . . . 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 2240 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))) |
| 42 | 1, 2, 3, 4, 5, 20, 29, 41 | climadd 15668 | 1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 + caddc 11158 ℤcz 12613 ℤ≥cuz 12878 ⇝ cli 15520 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 |
| This theorem is referenced by: fourierdlem112 46233 |
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