Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > xmulcand | Structured version Visualization version Unicode version |
Description: Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.) |
Ref | Expression |
---|---|
xmulcand.1 | |
xmulcand.2 | |
xmulcand.3 | |
xmulcand.4 |
Ref | Expression |
---|---|
xmulcand |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmulcand.3 | . . . 4 | |
2 | xmulcand.4 | . . . 4 | |
3 | xrecex 29628 | . . . 4 | |
4 | 1, 2, 3 | syl2anc 693 | . . 3 |
5 | oveq2 6658 | . . . 4 | |
6 | simprl 794 | . . . . . . . . . 10 | |
7 | 6 | rexrd 10089 | . . . . . . . . 9 |
8 | 1 | adantr 481 | . . . . . . . . . 10 |
9 | 8 | rexrd 10089 | . . . . . . . . 9 |
10 | xmulcom 12096 | . . . . . . . . 9 | |
11 | 7, 9, 10 | syl2anc 693 | . . . . . . . 8 |
12 | simprr 796 | . . . . . . . 8 | |
13 | 11, 12 | eqtrd 2656 | . . . . . . 7 |
14 | 13 | oveq1d 6665 | . . . . . 6 |
15 | xmulcand.1 | . . . . . . . 8 | |
16 | 15 | adantr 481 | . . . . . . 7 |
17 | xmulass 12117 | . . . . . . 7 | |
18 | 7, 9, 16, 17 | syl3anc 1326 | . . . . . 6 |
19 | xmulid2 12110 | . . . . . . 7 | |
20 | 16, 19 | syl 17 | . . . . . 6 |
21 | 14, 18, 20 | 3eqtr3d 2664 | . . . . 5 |
22 | 13 | oveq1d 6665 | . . . . . 6 |
23 | xmulcand.2 | . . . . . . . 8 | |
24 | 23 | adantr 481 | . . . . . . 7 |
25 | xmulass 12117 | . . . . . . 7 | |
26 | 7, 9, 24, 25 | syl3anc 1326 | . . . . . 6 |
27 | xmulid2 12110 | . . . . . . 7 | |
28 | 24, 27 | syl 17 | . . . . . 6 |
29 | 22, 26, 28 | 3eqtr3d 2664 | . . . . 5 |
30 | 21, 29 | eqeq12d 2637 | . . . 4 |
31 | 5, 30 | syl5ib 234 | . . 3 |
32 | 4, 31 | rexlimddv 3035 | . 2 |
33 | oveq2 6658 | . 2 | |
34 | 32, 33 | impbid1 215 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wrex 2913 (class class class)co 6650 cr 9935 cc0 9936 c1 9937 cxr 10073 cxmu 11945 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-xneg 11946 df-xmul 11948 |
This theorem is referenced by: xreceu 29630 |
Copyright terms: Public domain | W3C validator |