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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 24271 | . . 3 ⊢ 0𝑝 = (ℂ × {0}) | |
2 | 1 | fveq1i 6671 | . 2 ⊢ (0𝑝‘𝐴) = ((ℂ × {0})‘𝐴) |
3 | c0ex 10635 | . . 3 ⊢ 0 ∈ V | |
4 | 3 | fvconst2 6966 | . 2 ⊢ (𝐴 ∈ ℂ → ((ℂ × {0})‘𝐴) = 0) |
5 | 2, 4 | syl5eq 2868 | 1 ⊢ (𝐴 ∈ ℂ → (0𝑝‘𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {csn 4567 × cxp 5553 ‘cfv 6355 ℂcc 10535 0cc0 10537 0𝑝c0p 24270 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-0p 24271 |
This theorem is referenced by: 0plef 24273 0pledm 24274 itg1ge0 24287 mbfi1fseqlem5 24320 itg2addlem 24359 ne0p 24797 plyeq0lem 24800 plydivlem3 24884 plymul02 31816 dgraa0p 39769 |
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