Theorem sravsca 19182

Description: The scalar product operation of a subring algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
srapart.a	|- ( ph -> A = ( (subringAlg ` W ) ` S ) )
srapart.s	|- ( ph -> S C_ ( Base ` W ) )

Assertion

Ref	Expression
sravsca	|- ( ph -> ( .r ` W ) = ( .s ` A ) )

Proof of Theorem sravsca

Step	Hyp	Ref	Expression
1		srapart.a	. . . . . 6 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
2	1	adantl 482	. . . . 5 |- ( ( W e. _V /\ ph ) -> A = ( (subringAlg ` W ) ` S ) )
3		srapart.s	. . . . . 6 |- ( ph -> S C_ ( Base ` W ) )
4		sraval 19176	. . . . . 6 |- ( ( W e. _V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
5	3, 4	sylan2 491	. . . . 5 |- ( ( W e. _V /\ ph ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
6	2, 5	eqtrd 2656	. . . 4 |- ( ( W e. _V /\ ph ) -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
7	6	fveq2d 6195	. . 3 |- ( ( W e. _V /\ ph ) -> ( .s ` A ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
8		ovex 6678	. . . . 5 |- ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V
9		fvex 6201	. . . . 5 |- ( .r ` W ) e. _V
10		vscaid 16016	. . . . . 6 |- .s = Slot ( .s ` ndx )
11	10	setsid 15914	. . . . 5 |- ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) e. _V /\ ( .r ` W ) e. _V ) -> ( .r ` W ) = ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) )
12	8, 9, 11	mp2an 708	. . . 4 |- ( .r ` W ) = ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) )
13		6re 11101	. . . . . . 7 |- 6 e. RR
14		6lt8 11216	. . . . . . 7 |- 6 < 8
15	13, 14	ltneii 10150	. . . . . 6 |- 6 =/= 8
16		vscandx 16015	. . . . . . 7 |- ( .s ` ndx ) = 6
17		ipndx 16022	. . . . . . 7 |- ( .i ` ndx ) = 8
18	16, 17	neeq12i 2860	. . . . . 6 |- ( ( .s ` ndx ) =/= ( .i ` ndx ) <-> 6 =/= 8 )
19	15, 18	mpbir 221	. . . . 5 |- ( .s ` ndx ) =/= ( .i ` ndx )
20	10, 19	setsnid 15915	. . . 4 |- ( .s ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
21	12, 20	eqtri 2644	. . 3 |- ( .r ` W ) = ( .s ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
22	7, 21	syl6reqr 2675	. 2 |- ( ( W e. _V /\ ph ) -> ( .r ` W ) = ( .s ` A ) )
23	10	str0 15911	. . 3 |- (/) = ( .s ` (/) )
24		fvprc 6185	. . . 4 |- ( -. W e. _V -> ( .r ` W ) = (/) )
25	24	adantr 481	. . 3 |- ( ( -. W e. _V /\ ph ) -> ( .r ` W ) = (/) )
26		fvprc 6185	. . . . . . 7 |- ( -. W e. _V -> (subringAlg ` W ) = (/) )
27	26	fveq1d 6193	. . . . . 6 |- ( -. W e. _V -> ( (subringAlg ` W ) ` S ) = ( (/) ` S ) )
28		0fv 6227	. . . . . 6 |- ( (/) ` S ) = (/)
29	27, 28	syl6eq 2672	. . . . 5 |- ( -. W e. _V -> ( (subringAlg ` W ) ` S ) = (/) )
30	1, 29	sylan9eqr 2678	. . . 4 |- ( ( -. W e. _V /\ ph ) -> A = (/) )
31	30	fveq2d 6195	. . 3 |- ( ( -. W e. _V /\ ph ) -> ( .s ` A ) = ( .s ` (/) ) )
32	23, 25, 31	3eqtr4a 2682	. 2 |- ( ( -. W e. _V /\ ph ) -> ( .r ` W ) = ( .s ` A ) )
33	22, 32	pm2.61ian 831	1 |- ( ph -> ( .r ` W ) = ( .s ` A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   6c6 11074   8c8 11076   ndxcnx 15854   sSet csts 15855   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   .icip 15946  subringAlg csra 19168

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  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  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-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-sets 15864  df-vsca 15958  df-ip 15959  df-sra 19172

This theorem is referenced by:  sralmod  19187  rlmvsca  19202  sraassa  19325  sranlm  22488