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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 7382 | . . . 4 ⊢ 0 ∈ ℂ | |
2 | 1 | subidi 7655 | . . 3 ⊢ (0 − 0) = 0 |
3 | 2 | oveq1i 5600 | . 2 ⊢ ((0 − 0) · 𝐴) = (0 · 𝐴) |
4 | subdir 7766 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) | |
5 | 1, 1, 4 | mp3an12 1259 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = ((0 · 𝐴) − (0 · 𝐴))) |
6 | mulcl 7371 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (0 · 𝐴) ∈ ℂ) | |
7 | 6 | subidd 7683 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
8 | 1, 7 | mpan 415 | . . 3 ⊢ (𝐴 ∈ ℂ → ((0 · 𝐴) − (0 · 𝐴)) = 0) |
9 | 5, 8 | eqtrd 2115 | . 2 ⊢ (𝐴 ∈ ℂ → ((0 − 0) · 𝐴) = 0) |
10 | 3, 9 | syl5eqr 2129 | 1 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 (class class class)co 5590 ℂcc 7250 0cc0 7252 · cmul 7257 − cmin 7555 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-setind 4315 ax-resscn 7339 ax-1cn 7340 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-distr 7351 ax-i2m1 7352 ax-0id 7355 ax-rnegex 7356 ax-cnre 7358 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-sub 7557 |
This theorem is referenced by: mul02lem2 7768 mul01 7769 mul02i 7770 mul02d 7772 |
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