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Mirrors > Home > ILE Home > Th. List > mul02i | GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 23-Nov-1994.) |
Ref | Expression |
---|---|
mul01i.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
mul02i | ⊢ (0 · 𝐴) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01i.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mul02 8281 | . 2 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (0 · 𝐴) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1343 ∈ wcel 2136 (class class class)co 5841 ℂcc 7747 0cc0 7749 · cmul 7754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-14 2139 ax-ext 2147 ax-sep 4099 ax-pow 4152 ax-pr 4186 ax-setind 4513 ax-resscn 7841 ax-1cn 7842 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-distr 7853 ax-i2m1 7854 ax-0id 7857 ax-rnegex 7858 ax-cnre 7860 |
This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-ral 2448 df-rex 2449 df-reu 2450 df-rab 2452 df-v 2727 df-sbc 2951 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-br 3982 df-opab 4043 df-id 4270 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-iota 5152 df-fun 5189 df-fv 5195 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-sub 8067 |
This theorem is referenced by: sq10 10621 abs0 10996 odd2np1lem 11805 sin2pi 13324 |
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