Theorem opprmulfval 18625

Description: Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprval.1	|- B = ( Base ` R )
opprval.2	|- .x. = ( .r ` R )
opprval.3	|- O = (oppr ` R )
opprmulfval.4	|- .xb = ( .r ` O )

Assertion

Ref	Expression
opprmulfval	|- .xb = tpos .x.

Proof of Theorem opprmulfval

Step	Hyp	Ref	Expression
1		opprmulfval.4	. 2 |- .xb = ( .r ` O )
2		opprval.2	. . . . . . 7 |- .x. = ( .r ` R )
3		fvex 6201	. . . . . . 7 |- ( .r ` R ) e. _V
4	2, 3	eqeltri 2697	. . . . . 6 |- .x. e. _V
5	4	tposex 7386	. . . . 5 |- tpos .x. e. _V
6		mulrid 15997	. . . . . 6 |- .r = Slot ( .r ` ndx )
7	6	setsid 15914	. . . . 5 |- ( ( R e. _V /\ tpos .x. e. _V ) -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
8	5, 7	mpan2 707	. . . 4 |- ( R e. _V -> tpos .x. = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) ) )
9		opprval.1	. . . . . 6 |- B = ( Base ` R )
10		opprval.3	. . . . . 6 |- O = (oppr ` R )
11	9, 2, 10	opprval 18624	. . . . 5 |- O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. )
12	11	fveq2i 6194	. . . 4 |- ( .r ` O ) = ( .r ` ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
13	8, 12	syl6reqr 2675	. . 3 |- ( R e. _V -> ( .r ` O ) = tpos .x. )
14		tpos0 7382	. . . . 5 |- tpos (/) = (/)
15	6	str0 15911	. . . . 5 |- (/) = ( .r ` (/) )
16	14, 15	eqtr2i 2645	. . . 4 |- ( .r ` (/) ) = tpos (/)
17		fvprc 6185	. . . . . 6 |- ( -. R e. _V -> (oppr ` R ) = (/) )
18	10, 17	syl5eq 2668	. . . . 5 |- ( -. R e. _V -> O = (/) )
19	18	fveq2d 6195	. . . 4 |- ( -. R e. _V -> ( .r ` O ) = ( .r ` (/) ) )
20		fvprc 6185	. . . . . 6 |- ( -. R e. _V -> ( .r ` R ) = (/) )
21	2, 20	syl5eq 2668	. . . . 5 |- ( -. R e. _V -> .x. = (/) )
22	21	tposeqd 7355	. . . 4 |- ( -. R e. _V -> tpos .x. = tpos (/) )
23	16, 19, 22	3eqtr4a 2682	. . 3 |- ( -. R e. _V -> ( .r ` O ) = tpos .x. )
24	13, 23	pm2.61i 176	. 2 |- ( .r ` O ) = tpos .x.
25	1, 24	eqtri 2644	1 |- .xb = tpos .x.

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351   ndxcnx 15854   sSet csts 15855   Basecbs 15857   .rcmulr 15942  opp_rcoppr 18622

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-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009

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-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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-sets 15864  df-mulr 15955  df-oppr 18623

This theorem is referenced by:  opprmul  18626