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Mirrors > Home > ILE Home > Th. List > mulid1d | GIF version |
Description: Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
mulid1d | ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulid1 7910 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 1) = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 · 1) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5851 ℂcc 7765 1c1 7768 · cmul 7772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-resscn 7859 ax-1cn 7860 ax-icn 7862 ax-addcl 7863 ax-mulcl 7865 ax-mulcom 7868 ax-mulass 7870 ax-distr 7871 ax-1rid 7874 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5854 |
This theorem is referenced by: muladd11 8045 ltmul1 8504 mulap0 8565 divrecap 8598 diveqap1 8615 conjmulap 8639 apmul1 8698 qapne 9591 divelunit 9952 modqid 10298 q2submod 10334 addmodlteq 10347 expadd 10511 leexp2r 10523 nnlesq 10572 sqoddm1div8 10622 nn0opthlem1d 10647 faclbnd 10668 faclbnd2 10669 faclbnd6 10671 facavg 10673 bcn0 10682 bcn1 10685 reccn2ap 11269 hash2iun1dif1 11436 binom11 11442 trireciplem 11456 geosergap 11462 cvgratnnlemnexp 11480 cvgratnnlemmn 11481 fprodsplitdc 11552 efzval 11639 tanaddaplem 11694 tanaddap 11695 cos01gt0 11718 absef 11725 1dvds 11760 bezoutlema 11947 bezoutlemb 11948 gcdmultiple 11968 sqgcd 11977 lcm1 12028 coprmdvds 12039 qredeu 12044 phiprmpw 12169 coprimeprodsq 12204 pc2dvds 12276 sumhashdc 12292 fldivp1 12293 pcfaclem 12294 prmpwdvds 12300 dveflem 13446 efper 13487 tangtx 13518 logdivlti 13561 relogbexpap 13635 rplogbcxp 13640 lgsdir2 13693 2sqlem6 13715 2sqlem8 13718 trilpolemclim 14033 trilpolemisumle 14035 trilpolemeq1 14037 trilpolemlt1 14038 redcwlpolemeq1 14051 nconstwlpolemgt0 14060 |
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