Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ax1rid | Unicode version |
Description: is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7695. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax1rid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-r 7598 | . 2 | |
2 | oveq1 5749 | . . 3 | |
3 | id 19 | . . 3 | |
4 | 2, 3 | eqeq12d 2132 | . 2 |
5 | elsni 3515 | . . 3 | |
6 | df-1 7596 | . . . . . . 7 | |
7 | 6 | oveq2i 5753 | . . . . . 6 |
8 | 1sr 7527 | . . . . . . . 8 | |
9 | mulresr 7614 | . . . . . . . 8 | |
10 | 8, 9 | mpan2 421 | . . . . . . 7 |
11 | 1idsr 7544 | . . . . . . . 8 | |
12 | 11 | opeq1d 3681 | . . . . . . 7 |
13 | 10, 12 | eqtrd 2150 | . . . . . 6 |
14 | 7, 13 | syl5eq 2162 | . . . . 5 |
15 | opeq2 3676 | . . . . . . 7 | |
16 | 15 | oveq1d 5757 | . . . . . 6 |
17 | 16, 15 | eqeq12d 2132 | . . . . 5 |
18 | 14, 17 | syl5ibr 155 | . . . 4 |
19 | 18 | impcom 124 | . . 3 |
20 | 5, 19 | sylan2 284 | . 2 |
21 | 1, 4, 20 | optocl 4585 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1316 wcel 1465 csn 3497 cop 3500 (class class class)co 5742 cnr 7073 c0r 7074 c1r 7075 cmr 7078 cr 7587 c1 7589 cmul 7593 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-2o 6282 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-mpq 7121 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-mqqs 7126 df-1nqqs 7127 df-rq 7128 df-ltnqqs 7129 df-enq0 7200 df-nq0 7201 df-0nq0 7202 df-plq0 7203 df-mq0 7204 df-inp 7242 df-i1p 7243 df-iplp 7244 df-imp 7245 df-enr 7502 df-nr 7503 df-plr 7504 df-mr 7505 df-0r 7507 df-1r 7508 df-m1r 7509 df-c 7594 df-1 7596 df-r 7598 df-mul 7600 |
This theorem is referenced by: rereceu 7665 recriota 7666 |
Copyright terms: Public domain | W3C validator |