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Mirrors > Home > ILE Home > Th. List > 1idsr | Unicode version |
Description: 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.) |
Ref | Expression |
---|---|
1idsr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nr 6812 | . 2 | |
2 | oveq1 5519 | . . 3 | |
3 | id 19 | . . 3 | |
4 | 2, 3 | eqeq12d 2054 | . 2 |
5 | df-1r 6817 | . . . 4 | |
6 | 5 | oveq2i 5523 | . . 3 |
7 | 1pr 6652 | . . . . . 6 | |
8 | addclpr 6635 | . . . . . 6 | |
9 | 7, 7, 8 | mp2an 402 | . . . . 5 |
10 | mulsrpr 6831 | . . . . 5 | |
11 | 9, 7, 10 | mpanr12 415 | . . . 4 |
12 | distrprg 6686 | . . . . . . . . 9 | |
13 | 7, 7, 12 | mp3an23 1224 | . . . . . . . 8 |
14 | 1idpr 6690 | . . . . . . . . 9 | |
15 | 14 | oveq1d 5527 | . . . . . . . 8 |
16 | 13, 15 | eqtr2d 2073 | . . . . . . 7 |
17 | distrprg 6686 | . . . . . . . . 9 | |
18 | 7, 7, 17 | mp3an23 1224 | . . . . . . . 8 |
19 | 1idpr 6690 | . . . . . . . . 9 | |
20 | 19 | oveq1d 5527 | . . . . . . . 8 |
21 | 18, 20 | eqtrd 2072 | . . . . . . 7 |
22 | 16, 21 | oveqan12d 5531 | . . . . . 6 |
23 | simpl 102 | . . . . . . 7 | |
24 | mulclpr 6670 | . . . . . . . 8 | |
25 | 23, 7, 24 | sylancl 392 | . . . . . . 7 |
26 | mulclpr 6670 | . . . . . . . . 9 | |
27 | 9, 26 | mpan2 401 | . . . . . . . 8 |
28 | 27 | adantl 262 | . . . . . . 7 |
29 | addassprg 6677 | . . . . . . 7 | |
30 | 23, 25, 28, 29 | syl3anc 1135 | . . . . . 6 |
31 | mulclpr 6670 | . . . . . . . 8 | |
32 | 23, 9, 31 | sylancl 392 | . . . . . . 7 |
33 | simpr 103 | . . . . . . 7 | |
34 | mulclpr 6670 | . . . . . . . 8 | |
35 | 33, 7, 34 | sylancl 392 | . . . . . . 7 |
36 | addcomprg 6676 | . . . . . . . 8 | |
37 | 36 | adantl 262 | . . . . . . 7 |
38 | addassprg 6677 | . . . . . . . 8 | |
39 | 38 | adantl 262 | . . . . . . 7 |
40 | 32, 33, 35, 37, 39 | caov12d 5682 | . . . . . 6 |
41 | 22, 30, 40 | 3eqtr3d 2080 | . . . . 5 |
42 | 9, 31 | mpan2 401 | . . . . . . . . 9 |
43 | 7, 34 | mpan2 401 | . . . . . . . . 9 |
44 | addclpr 6635 | . . . . . . . . 9 | |
45 | 42, 43, 44 | syl2an 273 | . . . . . . . 8 |
46 | 7, 24 | mpan2 401 | . . . . . . . . 9 |
47 | addclpr 6635 | . . . . . . . . 9 | |
48 | 46, 27, 47 | syl2an 273 | . . . . . . . 8 |
49 | 45, 48 | anim12i 321 | . . . . . . 7 |
50 | enreceq 6821 | . . . . . . 7 | |
51 | 49, 50 | syldan 266 | . . . . . 6 |
52 | 51 | anidms 377 | . . . . 5 |
53 | 41, 52 | mpbird 156 | . . . 4 |
54 | 11, 53 | eqtr4d 2075 | . . 3 |
55 | 6, 54 | syl5eq 2084 | . 2 |
56 | 1, 4, 55 | ecoptocl 6193 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cop 3378 (class class class)co 5512 cec 6104 cnp 6389 c1p 6390 cpp 6391 cmp 6392 cer 6394 cnr 6395 c1r 6397 cmr 6400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-imp 6567 df-enr 6811 df-nr 6812 df-mr 6814 df-1r 6817 |
This theorem is referenced by: pn0sr 6856 axi2m1 6949 ax1rid 6951 axcnre 6955 |
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