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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 7786 | . 2 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (0 · 𝐴) = 0 |
Colors of variables: wff set class |
Syntax hints: = wceq 1287 ∈ wcel 1436 (class class class)co 5594 ℂcc 7269 0cc0 7271 · cmul 7276 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 ax-setind 4319 ax-resscn 7358 ax-1cn 7359 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-br 3815 df-opab 3869 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-iota 4937 df-fun 4974 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-sub 7576 |
This theorem is referenced by: sq10 9970 abs0 10332 odd2np1lem 10666 |
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