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Mirrors > Home > ILE Home > Th. List > mul02 | GIF version |
Description: Multiplication by 0. Theorem I.6 of [Apostol] p. 18. (Contributed by NM, 10-Aug-1999.) |
Ref | Expression |
---|---|
mul02 | ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7577 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | 1 | subidi 7850 | . . 3 ⊢ (0 − 0) = 0 |
3 | 2 | oveq1i 5700 | . 2 ⊢ ((0 − 0) · 𝐴) = (0 · 𝐴) |
4 | subdir 7961 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) | |
5 | 1, 1, 4 | mp3an12 1270 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) |
6 | mulcl 7566 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 · 𝐴) ∈ ℂ) | |
7 | 6 | subidd 7878 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
8 | 1, 7 | mpan 416 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
9 | 5, 8 | eqtrd 2127 | . 2 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = 0) |
10 | 3, 9 | syl5eqr 2141 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 (class class class)co 5690 ℂcc 7445 0cc0 7447 · cmul 7452 − cmin 7750 |
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: mul02lem2 7963 mul01 7964 mul02i 7965 mul02d 7967 demoivreALT 11228 |
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