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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 7763 | . 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 1331 ∈ wcel 1480 (class class class)co 5774 ℂcc 7618 1c1 7621 · cmul 7625 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-mulcl 7718 ax-mulcom 7721 ax-mulass 7723 ax-distr 7724 ax-1rid 7727 ax-cnre 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: muladd11 7895 ltmul1 8354 mulap0 8415 divrecap 8448 diveqap1 8465 conjmulap 8489 apmul1 8548 qapne 9431 divelunit 9785 modqid 10122 q2submod 10158 addmodlteq 10171 expadd 10335 leexp2r 10347 nnlesq 10396 sqoddm1div8 10444 nn0opthlem1d 10466 faclbnd 10487 faclbnd2 10488 faclbnd6 10490 facavg 10492 bcn0 10501 bcn1 10504 reccn2ap 11082 hash2iun1dif1 11249 binom11 11255 trireciplem 11269 geosergap 11275 cvgratnnlemnexp 11293 cvgratnnlemmn 11294 efzval 11389 tanaddaplem 11445 tanaddap 11446 cos01gt0 11469 absef 11476 1dvds 11507 bezoutlema 11687 bezoutlemb 11688 gcdmultiple 11708 sqgcd 11717 lcm1 11762 coprmdvds 11773 qredeu 11778 phiprmpw 11898 dveflem 12855 efper 12888 tangtx 12919 trilpolemclim 13229 trilpolemisumle 13231 trilpolemeq1 13233 trilpolemlt1 13234 |
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