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Mirrors > Home > MPE Home > Th. List > grpsubfval | Structured version Visualization version Unicode version |
Description: Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.) |
Ref | Expression |
---|---|
grpsubval.b | |
grpsubval.p | |
grpsubval.i | |
grpsubval.m |
Ref | Expression |
---|---|
grpsubfval |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubval.m | . . 3 | |
2 | fveq2 6191 | . . . . . 6 | |
3 | grpsubval.b | . . . . . 6 | |
4 | 2, 3 | syl6eqr 2674 | . . . . 5 |
5 | fveq2 6191 | . . . . . . 7 | |
6 | grpsubval.p | . . . . . . 7 | |
7 | 5, 6 | syl6eqr 2674 | . . . . . 6 |
8 | eqidd 2623 | . . . . . 6 | |
9 | fveq2 6191 | . . . . . . . 8 | |
10 | grpsubval.i | . . . . . . . 8 | |
11 | 9, 10 | syl6eqr 2674 | . . . . . . 7 |
12 | 11 | fveq1d 6193 | . . . . . 6 |
13 | 7, 8, 12 | oveq123d 6671 | . . . . 5 |
14 | 4, 4, 13 | mpt2eq123dv 6717 | . . . 4 |
15 | df-sbg 17427 | . . . 4 | |
16 | fvex 6201 | . . . . . 6 | |
17 | 3, 16 | eqeltri 2697 | . . . . 5 |
18 | 17, 17 | mpt2ex 7247 | . . . 4 |
19 | 14, 15, 18 | fvmpt 6282 | . . 3 |
20 | 1, 19 | syl5eq 2668 | . 2 |
21 | fvprc 6185 | . . . 4 | |
22 | 1, 21 | syl5eq 2668 | . . 3 |
23 | fvprc 6185 | . . . . . 6 | |
24 | 3, 23 | syl5eq 2668 | . . . . 5 |
25 | mpt2eq12 6715 | . . . . 5 | |
26 | 24, 24, 25 | syl2anc 693 | . . . 4 |
27 | mpt20 6725 | . . . 4 | |
28 | 26, 27 | syl6eq 2672 | . . 3 |
29 | 22, 28 | eqtr4d 2659 | . 2 |
30 | 20, 29 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1483 wcel 1990 cvv 3200 c0 3915 cfv 5888 (class class class)co 6650 cmpt2 6652 cbs 15857 cplusg 15941 cminusg 17423 csg 17424 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 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-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-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-sbg 17427 |
This theorem is referenced by: grpsubval 17465 grpsubf 17494 grpsubpropd 17520 grpsubpropd2 17521 tgpsubcn 21894 tngtopn 22454 |
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