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Mirrors > Home > ILE Home > Th. List > mulid1i | GIF version |
Description: Identity law for multiplication. (Contributed by NM, 14-Feb-1995.) |
Ref | Expression |
---|---|
axi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
mulid1i | ⊢ (𝐴 · 1) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | mulid1 7942 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 · 1) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5869 ℂcc 7797 1c1 7800 · cmul 7804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-resscn 7891 ax-1cn 7892 ax-icn 7894 ax-addcl 7895 ax-mulcl 7897 ax-mulcom 7900 ax-mulass 7902 ax-distr 7903 ax-1rid 7906 ax-cnre 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 |
This theorem is referenced by: rimul 8529 muleqadd 8611 1t1e1 9057 2t1e2 9058 3t1e3 9060 halfpm6th 9125 iap0 9128 9p1e10 9372 numltc 9395 numsucc 9409 dec10p 9412 numadd 9416 numaddc 9417 11multnc 9437 4t3lem 9466 5t2e10 9469 9t11e99 9499 rei 10889 imi 10890 cji 10892 0.999... 11510 efival 11721 ef01bndlem 11745 3lcm2e6 12140 dveflem 13851 efhalfpi 13884 |
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