Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23243 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | 1 | fveq1i 6104 | . 2 ⊢ (0𝑝‘𝐴) = ((ℂ × {0})‘𝐴) |
3 | c0ex 9913 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 6374 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0) |
5 | 2, 4 | syl5eq 2656 | 1 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 × cxp 5036 ‘cfv 5804 ℂcc 9813 0cc0 9815 0𝑝c0p 23242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-0p 23243 |
This theorem is referenced by: 0plef 23245 0pledm 23246 itg1ge0 23259 mbfi1fseqlem5 23292 itg2addlem 23331 ne0p 23767 plyeq0lem 23770 plydivlem3 23854 plymul02 29949 dgraa0p 36738 |
Copyright terms: Public domain | W3C validator |