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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 7787 | . 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 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 1c1 7645 · cmul 7649 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-mulcl 7742 ax-mulcom 7745 ax-mulass 7747 ax-distr 7748 ax-1rid 7751 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 |
This theorem is referenced by: muladd11 7919 ltmul1 8378 mulap0 8439 divrecap 8472 diveqap1 8489 conjmulap 8513 apmul1 8572 qapne 9458 divelunit 9815 modqid 10153 q2submod 10189 addmodlteq 10202 expadd 10366 leexp2r 10378 nnlesq 10427 sqoddm1div8 10475 nn0opthlem1d 10498 faclbnd 10519 faclbnd2 10520 faclbnd6 10522 facavg 10524 bcn0 10533 bcn1 10536 reccn2ap 11114 hash2iun1dif1 11281 binom11 11287 trireciplem 11301 geosergap 11307 cvgratnnlemnexp 11325 cvgratnnlemmn 11326 efzval 11426 tanaddaplem 11481 tanaddap 11482 cos01gt0 11505 absef 11512 1dvds 11543 bezoutlema 11723 bezoutlemb 11724 gcdmultiple 11744 sqgcd 11753 lcm1 11798 coprmdvds 11809 qredeu 11814 phiprmpw 11934 dveflem 12895 efper 12936 tangtx 12967 logdivlti 13010 relogbexpap 13083 rplogbcxp 13088 trilpolemclim 13404 trilpolemisumle 13406 trilpolemeq1 13408 trilpolemlt1 13409 redcwlpolemeq1 13421 |
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