Theorem imasmulr 13112

Description: The ring multiplication in an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
imasbas.u	|- ( ph -> U = ( F "s R ) )
imasbas.v	|- ( ph -> V = ( Base ` R ) )
imasbas.f	|- ( ph -> F : V -onto-> B )
imasbas.r	|- ( ph -> R e. Z )
imasmulr.p	|- .x. = ( .r ` R )
imasmulr.t	|- .xb = ( .r ` U )

Assertion

Ref	Expression
imasmulr	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )

Distinct variable groups:   F, p, q   R, p, q   V, p, q   ph, p, q

Allowed substitution hints:   B( q, p)   .xb ( q, p)   .x. ( q, p)   U( q, p)   Z( q, p)

Proof of Theorem imasmulr

Step	Hyp	Ref	Expression
1		imasmulr.t	. 2 |- .xb = ( .r ` U )
2		imasbas.u	. . . . 5 |- ( ph -> U = ( F "s R ) )
3		imasbas.v	. . . . 5 |- ( ph -> V = ( Base ` R ) )
4		eqid 2204	. . . . 5 |- ( +g ` R ) = ( +g ` R )
5		imasmulr.p	. . . . 5 |- .x. = ( .r ` R )
6		eqid 2204	. . . . 5 |- ( .s ` R ) = ( .s ` R )
7		eqidd 2205	. . . . 5 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
8		eqidd 2205	. . . . 5 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
9		imasbas.f	. . . . 5 |- ( ph -> F : V -onto-> B )
10		imasbas.r	. . . . 5 |- ( ph -> R e. Z )
11	2, 3, 4, 5, 6, 7, 8, 9, 10	imasival 13109	. . . 4 |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } >. } )
12	11	fveq1d 5577	. . 3 |- ( ph -> ( U ` ( .r ` ndx ) ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } >. } ` ( .r ` ndx ) ) )
13		fof 5497	. . . . . . . 8 |- ( F : V -onto-> B -> F : V --> B )
14	9, 13	syl 14	. . . . . . 7 |- ( ph -> F : V --> B )
15		basfn 12861	. . . . . . . . 9 |- Base Fn _V
16	10	elexd 2784	. . . . . . . . 9 |- ( ph -> R e. _V )
17		funfvex 5592	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5375	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
19	15, 16, 18	sylancr 414	. . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
20	3, 19	eqeltrd 2281	. . . . . . 7 |- ( ph -> V e. _V )
21	14, 20	fexd 5813	. . . . . 6 |- ( ph -> F e. _V )
22		imasex 13108	. . . . . 6 |- ( ( F e. _V /\ R e. Z ) -> ( F "s R ) e. _V )
23	21, 10, 22	syl2anc 411	. . . . 5 |- ( ph -> ( F "s R ) e. _V )
24	2, 23	eqeltrd 2281	. . . 4 |- ( ph -> U e. _V )
25		mulridx 12934	. . . 4 |- .r = Slot ( .r ` ndx )
26		mulrslid 12935	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . 4 |- ( .r ` ndx ) e. NN
28	24, 25, 27	strndxid 12831	. . 3 |- ( ph -> ( U ` ( .r ` ndx ) ) = ( .r ` U ) )
29	27	a1i 9	. . . 4 |- ( ph -> ( .r ` ndx ) e. NN )
30		vex 2774	. . . . . . . . . . . 12 |- p e. _V
31		fvexg 5594	. . . . . . . . . . . 12 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
32	21, 30, 31	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( F ` p ) e. _V )
33		vex 2774	. . . . . . . . . . . 12 |- q e. _V
34		fvexg 5594	. . . . . . . . . . . 12 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
35	21, 33, 34	sylancl 413	. . . . . . . . . . 11 |- ( ph -> ( F ` q ) e. _V )
36		opexg 4271	. . . . . . . . . . 11 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
37	32, 35, 36	syl2anc 411	. . . . . . . . . 10 |- ( ph -> <. ( F ` p ) , ( F ` q ) >. e. _V )
38	26	slotex 12830	. . . . . . . . . . . . . 14 |- ( R e. Z -> ( .r ` R ) e. _V )
39	10, 38	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( .r ` R ) e. _V )
40	5, 39	eqeltrid 2291	. . . . . . . . . . . 12 |- ( ph -> .x. e. _V )
41	33	a1i 9	. . . . . . . . . . . 12 |- ( ph -> q e. _V )
42		ovexg 5977	. . . . . . . . . . . 12 |- ( ( p e. _V /\ .x. e. _V /\ q e. _V ) -> ( p .x. q ) e. _V )
43	30, 40, 41, 42	mp3an2i 1354	. . . . . . . . . . 11 |- ( ph -> ( p .x. q ) e. _V )
44		fvexg 5594	. . . . . . . . . . 11 |- ( ( F e. _V /\ ( p .x. q ) e. _V ) -> ( F ` ( p .x. q ) ) e. _V )
45	21, 43, 44	syl2anc 411	. . . . . . . . . 10 |- ( ph -> ( F ` ( p .x. q ) ) e. _V )
46		opexg 4271	. . . . . . . . . 10 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p .x. q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. _V )
47	37, 45, 46	syl2anc 411	. . . . . . . . 9 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. _V )
48		snexg 4227	. . . . . . . . 9 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
49	47, 48	syl 14	. . . . . . . 8 |- ( ph -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
50	49	ralrimivw 2579	. . . . . . 7 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
51		iunexg 6203	. . . . . . 7 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
52	20, 50, 51	syl2anc 411	. . . . . 6 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
53	52	ralrimivw 2579	. . . . 5 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
54		iunexg 6203	. . . . 5 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
55	20, 53, 54	syl2anc 411	. . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
56		basendxnmulrndx 12937	. . . . 5 |- ( Base ` ndx ) =/= ( .r ` ndx )
57	56	a1i 9	. . . 4 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
58		plusgndxnmulrndx 12936	. . . . 5 |- ( +g ` ndx ) =/= ( .r ` ndx )
59	58	a1i 9	. . . 4 |- ( ph -> ( +g ` ndx ) =/= ( .r ` ndx ) )
60		fvtp3g 5793	. . . 4 |- ( ( ( ( .r ` ndx ) e. NN /\ U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V ) /\ ( ( Base ` ndx ) =/= ( .r ` ndx ) /\ ( +g ` ndx ) =/= ( .r ` ndx ) ) ) -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } >. } ` ( .r ` ndx ) ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
61	29, 55, 57, 59, 60	syl22anc 1250	. . 3 |- ( ph -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } >. } ` ( .r ` ndx ) ) = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )
62	12, 28, 61	3eqtr3rd 2246	. 2 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } = ( .r ` U ) )
63	1, 62	eqtr4id 2256	1 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   =/= wne 2375   A.wral 2483   _Vcvv 2771   {csn 3632   {ctp 3634   <.cop 3635   U_ciun 3926   Fn wfn 5265   -->wf 5266   -onto->wfo 5268   ` cfv 5270  (class class class)co 5943   NNcn 9035   ndxcnx 12800  Slot cslot 12802   Basecbs 12803   +g cplusg 12880   .rcmulr 12881   .scvsca 12884   "_s cimas 13102

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-pre-ltirr 8036  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-tp 3640  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-pnf 8108  df-mnf 8109  df-ltxr 8111  df-inn 9036  df-2 9094  df-3 9095  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-mulr 12894  df-iimas 13105

This theorem is referenced by:  imasmulfn  13123  imasmulval  13124  imasmulf  13125  qusmulval  13140  qusmulf  13141