Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7917 | . 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 5853 ℂcc 7772 1c1 7775 · cmul 7779 |
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 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-mulcl 7872 ax-mulcom 7875 ax-mulass 7877 ax-distr 7878 ax-1rid 7881 ax-cnre 7885 |
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 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 |
This theorem is referenced by: muladd11 8052 ltmul1 8511 mulap0 8572 divrecap 8605 diveqap1 8622 conjmulap 8646 apmul1 8705 qapne 9598 divelunit 9959 modqid 10305 q2submod 10341 addmodlteq 10354 expadd 10518 leexp2r 10530 nnlesq 10579 sqoddm1div8 10629 nn0opthlem1d 10654 faclbnd 10675 faclbnd2 10676 faclbnd6 10678 facavg 10680 bcn0 10689 bcn1 10692 reccn2ap 11276 hash2iun1dif1 11443 binom11 11449 trireciplem 11463 geosergap 11469 cvgratnnlemnexp 11487 cvgratnnlemmn 11488 fprodsplitdc 11559 efzval 11646 tanaddaplem 11701 tanaddap 11702 cos01gt0 11725 absef 11732 1dvds 11767 bezoutlema 11954 bezoutlemb 11955 gcdmultiple 11975 sqgcd 11984 lcm1 12035 coprmdvds 12046 qredeu 12051 phiprmpw 12176 coprimeprodsq 12211 pc2dvds 12283 sumhashdc 12299 fldivp1 12300 pcfaclem 12301 prmpwdvds 12307 dveflem 13481 efper 13522 tangtx 13553 logdivlti 13596 relogbexpap 13670 rplogbcxp 13675 lgsdir2 13728 2sqlem6 13750 2sqlem8 13753 trilpolemclim 14068 trilpolemisumle 14070 trilpolemeq1 14072 trilpolemlt1 14073 redcwlpolemeq1 14086 nconstwlpolemgt0 14095 |
Copyright terms: Public domain | W3C validator |