Theorem qusmulval 13254

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 2206	. 2 |- ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] .~ )
8		basfn 12975	. . . . . . 7 |- Base Fn _V
9	4	elexd 2787	. . . . . . 7 |- ( ph -> R e. _V )
10		funfvex 5611	. . . . . . . 8 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
11	10	funfni 5390	. . . . . . 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 2283	. . . . 5 |- ( ph -> V e. _V )
14		erex 6662	. . . . 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 13240	. . 3 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
17	1, 2, 7, 15, 4	quslem 13241	. . 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 13226	. 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 13049	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
22	21	slotex 12944	. . . 4 |- ( R e. Z -> ( .r ` R ) e. _V )
23	4, 22	syl 14	. . 3 |- ( ph -> ( .r ` R ) e. _V )
24	18, 23	eqeltrid 2293	. 2 |- ( ph -> .x. e. _V )
25	1, 2, 3, 4, 5, 6, 7, 20, 24	qusaddvallemg 13250	1 |- ( ( ph /\ X e. V /\ Y e. V ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 981   = wceq 1373   e. wcel 2177   _Vcvv 2773   class class class wbr 4054   |-> cmpt 4116   Fn wfn 5280   ` cfv 5285  (class class class)co 5962   Er wer 6635   [cec 6636   /.cqs 6637   Basecbs 12917   .rcmulr 12995   /._s cqus 13217

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-er 6638  df-ec 6640  df-qs 6644  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-plusg 13007  df-mulr 13008  df-iimas 13219  df-qus 13220

This theorem is referenced by:  qusrhm  14375  qusmul2  14376  qusmulrng  14379