Theorem qusmulval 13365

Description: The multiplication in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
qusaddf.u	|- ( ph -> U = ( R /.s .~ ) )
qusaddf.v	|- ( ph -> V = ( Base ` R ) )
qusaddf.r	|- ( ph -> .~ Er V )
qusaddf.z	|- ( ph -> R e. Z )
qusaddf.e	|- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
qusaddf.c	|- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
qusmulf.p	|- .x. = ( .r ` R )
qusmulf.a	|- .xb = ( .r ` U )

Assertion

Ref	Expression
qusmulval	|- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Distinct variable groups:   a, b, p, q, .~   ph, a, b, p, q   V, a, b, p, q   R, p, q   .x. , p, q   X, p, q   .xb , a, b, p, q   Y, p, q

Allowed substitution hints:   R( a, b)   .x. ( a, b)   U( q, p, a, b)   X( a, b)   Y( a, b)   Z( q, p, a, b)

Proof of Theorem qusmulval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusaddf.u	. 2 |- ( ph -> U = ( R /.s .~ ) )
2		qusaddf.v	. 2 |- ( ph -> V = ( Base ` R ) )
3		qusaddf.r	. 2 |- ( ph -> .~ Er V )
4		qusaddf.z	. 2 |- ( ph -> R e. Z )
5		qusaddf.e	. 2 |- ( ph -> ( ( a .~ p /\ b .~ q ) -> ( a .x. b ) .~ ( p .x. q ) ) )
6		qusaddf.c	. 2 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> ( p .x. q ) e. V )
7		eqid 2229	. 2 |- ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] .~ )
8		basfn 13086	. . . . . . 7 |- Base Fn _V
9	4	elexd 2813	. . . . . . 7 |- ( ph -> R e. _V )
10		funfvex 5643	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
11	10	funfni 5422	. . . . . . 7 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
12	8, 9, 11	sylancr 414	. . . . . 6 |- ( ph -> ( Base ` R ) e. _V )
13	2, 12	eqeltrd 2306	. . . . 5 |- ( ph -> V e. _V )
14		erex 6702	. . . . 5 |- ( .~ Er V -> ( V e. _V -> .~ e. _V ) )
15	3, 13, 14	sylc 62	. . . 4 |- ( ph -> .~ e. _V )
16	1, 2, 7, 15, 4	qusval 13351	. . 3 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
17	1, 2, 7, 15, 4	quslem 13352	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) : V -onto-> ( V /. .~ ) )
18		qusmulf.p	. . 3 |- .x. = ( .r ` R )
19		qusmulf.a	. . 3 |- .xb = ( .r ` U )
20	16, 2, 17, 4, 18, 19	imasmulr 13337	. 2 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( ( x e. V |-> [ x ] .~ ) ` p ) , ( ( x e. V |-> [ x ] .~ ) ` q ) >. , ( ( x e. V |-> [ x ] .~ ) ` ( p .x. q ) ) >. } )
21		mulrslid 13160	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 13054	. . . 4 |- ( R e. Z -> ( .r ` R ) e. _V )
23	4, 22	syl 14	. . 3 |- ( ph -> ( .r ` R ) e. _V )
24	18, 23	eqeltrid 2316	. 2 |- ( ph -> .x. e. _V )
25	1, 2, 3, 4, 5, 6, 7, 20, 24	qusaddvallemg 13361	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2799   class class class wbr 4082   |-> cmpt 4144   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   Er wer 6675   [cec 6676   /.cqs 6677   Basecbs 13027   .rcmulr 13106   /._s cqus 13328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-lttrn 8109  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-er 6678  df-ec 6680  df-qs 6684  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-plusg 13118  df-mulr 13119  df-iimas 13330  df-qus 13331

This theorem is referenced by:  qusrhm  14486  qusmul2  14487  qusmulrng  14490