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Mirrors > Home > MPE Home > Th. List > mulid1 | Structured version Visualization version Unicode version |
Description: is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mulid1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnre 10036 | . 2 | |
2 | recn 10026 | . . . . . 6 | |
3 | ax-icn 9995 | . . . . . . 7 | |
4 | recn 10026 | . . . . . . 7 | |
5 | mulcl 10020 | . . . . . . 7 | |
6 | 3, 4, 5 | sylancr 695 | . . . . . 6 |
7 | ax-1cn 9994 | . . . . . . 7 | |
8 | adddir 10031 | . . . . . . 7 | |
9 | 7, 8 | mp3an3 1413 | . . . . . 6 |
10 | 2, 6, 9 | syl2an 494 | . . . . 5 |
11 | ax-1rid 10006 | . . . . . 6 | |
12 | mulass 10024 | . . . . . . . . 9 | |
13 | 3, 7, 12 | mp3an13 1415 | . . . . . . . 8 |
14 | 4, 13 | syl 17 | . . . . . . 7 |
15 | ax-1rid 10006 | . . . . . . . 8 | |
16 | 15 | oveq2d 6666 | . . . . . . 7 |
17 | 14, 16 | eqtrd 2656 | . . . . . 6 |
18 | 11, 17 | oveqan12d 6669 | . . . . 5 |
19 | 10, 18 | eqtrd 2656 | . . . 4 |
20 | oveq1 6657 | . . . . 5 | |
21 | id 22 | . . . . 5 | |
22 | 20, 21 | eqeq12d 2637 | . . . 4 |
23 | 19, 22 | syl5ibrcom 237 | . . 3 |
24 | 23 | rexlimivv 3036 | . 2 |
25 | 1, 24 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wrex 2913 (class class class)co 6650 cc 9934 cr 9935 c1 9937 ci 9938 caddc 9939 cmul 9941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 ax-distr 10003 ax-1rid 10006 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: mulid2 10038 mulid1i 10042 mulid1d 10057 muleqadd 10671 divdiv1 10736 conjmul 10742 nnmulcl 11043 expmul 12905 binom21 12980 binom2sub1 12982 sq01 12986 bernneq 12990 hashiun 14554 fprodcvg 14660 prodmolem2a 14664 efexp 14831 cncrng 19767 cnfld1 19771 0dgr 24001 ecxp 24419 dvcxp1 24481 dvcncxp1 24484 efrlim 24696 lgsdilem2 25058 axcontlem7 25850 ipasslem2 27687 addltmulALT 29305 0dp2dp 29617 zrhnm 30013 2even 41933 |
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