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Mirrors > Home > ILE Home > Th. List > negfcncf | GIF version |
Description: The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
negfcncf.1 | ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥)) |
Ref | Expression |
---|---|
negfcncf | ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncff 14000 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹:𝐴⟶ℂ) | |
2 | 1 | ffvelcdmda 5651 | . . . 4 ⊢ ((𝐹 ∈ (𝐴–cn→ℂ) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ℂ) |
3 | 1 | feqmptd 5569 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
4 | eqidd 2178 | . . . 4 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → (𝑦 ∈ ℂ ↦ -𝑦) = (𝑦 ∈ ℂ ↦ -𝑦)) | |
5 | negeq 8149 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → -𝑦 = -(𝐹‘𝑥)) | |
6 | 2, 3, 4, 5 | fmptco 5682 | . . 3 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → ((𝑦 ∈ ℂ ↦ -𝑦) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥))) |
7 | negfcncf.1 | . . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ -(𝐹‘𝑥)) | |
8 | 6, 7 | eqtr4di 2228 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → ((𝑦 ∈ ℂ ↦ -𝑦) ∘ 𝐹) = 𝐺) |
9 | id 19 | . . 3 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐹 ∈ (𝐴–cn→ℂ)) | |
10 | ssid 3175 | . . . 4 ⊢ ℂ ⊆ ℂ | |
11 | eqid 2177 | . . . . 5 ⊢ (𝑦 ∈ ℂ ↦ -𝑦) = (𝑦 ∈ ℂ ↦ -𝑦) | |
12 | 11 | negcncf 14024 | . . . 4 ⊢ (ℂ ⊆ ℂ → (𝑦 ∈ ℂ ↦ -𝑦) ∈ (ℂ–cn→ℂ)) |
13 | 10, 12 | mp1i 10 | . . 3 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → (𝑦 ∈ ℂ ↦ -𝑦) ∈ (ℂ–cn→ℂ)) |
14 | 9, 13 | cncfco 14014 | . 2 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → ((𝑦 ∈ ℂ ↦ -𝑦) ∘ 𝐹) ∈ (𝐴–cn→ℂ)) |
15 | 8, 14 | eqeltrrd 2255 | 1 ⊢ (𝐹 ∈ (𝐴–cn→ℂ) → 𝐺 ∈ (𝐴–cn→ℂ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 ↦ cmpt 4064 ∘ ccom 4630 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 -cneg 8128 –cn→ccncf 13993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-map 6649 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-2 8977 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 df-cncf 13994 |
This theorem is referenced by: ivthdec 14058 |
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