Theorem qusmulf 13551

Description: The multiplication in a quotient structure as a function. (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
qusmulf	|- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )

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

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

Proof of Theorem qusmulf

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 2232	. 2 |- ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] .~ )
8		basfn 13271	. . . . . . 7 |- Base Fn _V
9	4	elexd 2827	. . . . . . 7 |- ( ph -> R e. _V )
10		funfvex 5687	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
11	10	funfni 5458	. . . . . . 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 2309	. . . . 5 |- ( ph -> V e. _V )
14		erex 6791	. . . . 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 13536	. . 3 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
17	1, 2, 7, 15, 4	quslem 13537	. . 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 13522	. 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 13345	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 13239	. . . 4 |- ( R e. Z -> ( .r ` R ) e. _V )
23	4, 22	syl 14	. . 3 |- ( ph -> ( .r ` R ) e. _V )
24	18, 23	eqeltrid 2319	. 2 |- ( ph -> .x. e. _V )
25	1, 2, 3, 4, 5, 6, 7, 20, 24	qusaddflemg 13547	1 |- ( ph -> .xb : ( ( V /. .~ ) X. ( V /. .~ ) ) --> ( V /. .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   |-> cmpt 4171   X. cxp 4747   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   Er wer 6764   [cec 6765   /.cqs 6766   Basecbs 13212   .rcmulr 13291   /._s cqus 13513

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-er 6767  df-ec 6769  df-qs 6773  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-iimas 13515  df-qus 13516

This theorem is referenced by: (None)