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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 24273 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | 1 | fveq1i 6673 | . 2 ⊢ (0𝑝‘𝐴) = ((ℂ × {0})‘𝐴) |
3 | c0ex 10637 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 6968 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0) |
5 | 2, 4 | syl5eq 2870 | 1 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4569 × cxp 5555 ‘cfv 6357 ℂcc 10537 0cc0 10539 0𝑝c0p 24272 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-mulcl 10601 ax-i2m1 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-0p 24273 |
This theorem is referenced by: 0plef 24275 0pledm 24276 itg1ge0 24289 mbfi1fseqlem5 24322 itg2addlem 24361 ne0p 24799 plyeq0lem 24802 plydivlem3 24886 plymul02 31818 dgraa0p 39756 |
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