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Mirrors > Home > ILE Home > Th. List > mul02d | GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mul01d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
mul02d | ⊢ (𝜑 → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mul02 7962 | . 2 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∈ wcel 1445 (class class class)co 5690 ℂcc 7445 0cc0 7447 · cmul 7452 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-setind 4381 ax-resscn 7534 ax-1cn 7535 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-sub 7752 |
This theorem is referenced by: mulneg1 7970 mulap0r 8189 mulap0 8220 un0mulcl 8805 lincmb01cmp 9569 iccf1o 9570 bcval5 10286 hashxp 10349 remul2 10422 immul2 10429 fsumconst 10997 binomlem 11026 efne0 11117 dvds0 11238 mulmoddvds 11291 mulgcd 11432 bezoutr1 11449 lcmgcd 11487 qnumgt0 11603 |
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