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Mirrors > Home > MPE Home > Th. List > 0pval | Structured version Visualization version GIF version |
Description: The zero function evaluates to zero at every point. (Contributed by Mario Carneiro, 23-Jul-2014.) |
Ref | Expression |
---|---|
0pval | ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0p 24274 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | 1 | fveq1i 6646 | . 2 ⊢ (0𝑝‘𝐴) = ((ℂ × {0})‘𝐴) |
3 | c0ex 10624 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 6943 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0) |
5 | 2, 4 | syl5eq 2845 | 1 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {csn 4525 × cxp 5517 ‘cfv 6324 ℂcc 10524 0cc0 10526 0𝑝c0p 24273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-0p 24274 |
This theorem is referenced by: 0plef 24276 0pledm 24277 itg1ge0 24290 mbfi1fseqlem5 24323 itg2addlem 24362 ne0p 24804 plyeq0lem 24807 plydivlem3 24891 plymul02 31926 dgraa0p 40093 |
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