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Mirrors > Home > MPE Home > Th. List > mulgfvi | Structured version Visualization version GIF version |
Description: The group multiple operation is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
mulgfvi.t | โข ยท = (.gโ๐บ) |
Ref | Expression |
---|---|
mulgfvi | โข ยท = (.gโ( I โ๐บ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulgfvi.t | . 2 โข ยท = (.gโ๐บ) | |
2 | fvi 6979 | . . . . 5 โข (๐บ โ V โ ( I โ๐บ) = ๐บ) | |
3 | 2 | eqcomd 2734 | . . . 4 โข (๐บ โ V โ ๐บ = ( I โ๐บ)) |
4 | 3 | fveq2d 6906 | . . 3 โข (๐บ โ V โ (.gโ๐บ) = (.gโ( I โ๐บ))) |
5 | fvprc 6894 | . . . 4 โข (ยฌ ๐บ โ V โ (.gโ๐บ) = โ ) | |
6 | fvprc 6894 | . . . . . 6 โข (ยฌ ๐บ โ V โ ( I โ๐บ) = โ ) | |
7 | 6 | fveq2d 6906 | . . . . 5 โข (ยฌ ๐บ โ V โ (.gโ( I โ๐บ)) = (.gโโ )) |
8 | base0 17192 | . . . . . . . 8 โข โ = (Baseโโ ) | |
9 | eqid 2728 | . . . . . . . 8 โข (.gโโ ) = (.gโโ ) | |
10 | 8, 9 | mulgfn 19035 | . . . . . . 7 โข (.gโโ ) Fn (โค ร โ ) |
11 | xp0 6167 | . . . . . . . 8 โข (โค ร โ ) = โ | |
12 | 11 | fneq2i 6657 | . . . . . . 7 โข ((.gโโ ) Fn (โค ร โ ) โ (.gโโ ) Fn โ ) |
13 | 10, 12 | mpbi 229 | . . . . . 6 โข (.gโโ ) Fn โ |
14 | fn0 6691 | . . . . . 6 โข ((.gโโ ) Fn โ โ (.gโโ ) = โ ) | |
15 | 13, 14 | mpbi 229 | . . . . 5 โข (.gโโ ) = โ |
16 | 7, 15 | eqtrdi 2784 | . . . 4 โข (ยฌ ๐บ โ V โ (.gโ( I โ๐บ)) = โ ) |
17 | 5, 16 | eqtr4d 2771 | . . 3 โข (ยฌ ๐บ โ V โ (.gโ๐บ) = (.gโ( I โ๐บ))) |
18 | 4, 17 | pm2.61i 182 | . 2 โข (.gโ๐บ) = (.gโ( I โ๐บ)) |
19 | 1, 18 | eqtri 2756 | 1 โข ยท = (.gโ( I โ๐บ)) |
Colors of variables: wff setvar class |
Syntax hints: ยฌ wn 3 = wceq 1533 โ wcel 2098 Vcvv 3473 โ c0 4326 I cid 5579 ร cxp 5680 Fn wfn 6548 โcfv 6553 โคcz 12596 .gcmg 19030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-seq 14007 df-slot 17158 df-ndx 17170 df-base 17188 df-mulg 19031 |
This theorem is referenced by: (None) |
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