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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 25687 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | 1 | fveq1i 6894 | . 2 ⊢ (0𝑝‘𝐴) = ((ℂ × {0})‘𝐴) |
3 | c0ex 11249 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 7213 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0) |
5 | 2, 4 | eqtrid 2778 | 1 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {csn 4623 × cxp 5672 ‘cfv 6546 ℂcc 11147 0cc0 11149 0𝑝c0p 25686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-mulcl 11211 ax-i2m1 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-0p 25687 |
This theorem is referenced by: 0plef 25689 0pledm 25690 itg1ge0 25703 mbfi1fseqlem5 25737 itg2addlem 25776 ne0p 26231 plyeq0lem 26234 plydivlem3 26320 plymul02 34405 dgraa0p 42847 |
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