Theorem imasmulr 13607

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 2238	. . . . 5 |- ( +g ` R ) = ( +g ` R )
5		imasmulr.p	. . . . 5 |- .x. = ( .r ` R )
6		eqid 2238	. . . . 5 |- ( .s ` R ) = ( .s ` R )
7		eqidd 2239	. . . . 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 2239	. . . . 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 13604	. . . 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 5692	. . 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 5610	. . . . . . . 8 |- ( F : V -onto-> B -> F : V --> B )
14	9, 13	syl 14	. . . . . . 7 |- ( ph -> F : V --> B )
15		basfn 13389	. . . . . . . . 9 |- Base Fn _V
16	10	elexd 2835	. . . . . . . . 9 |- ( ph -> R e. _V )
17		funfvex 5707	. . . . . . . . . 10 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
18	17	funfni 5478	. . . . . . . . 9 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
19	15, 16, 18	sylancr 418	. . . . . . . 8 |- ( ph -> ( Base ` R ) e. _V )
20	3, 19	eqeltrd 2315	. . . . . . 7 |- ( ph -> V e. _V )
21	14, 20	fexd 5938	. . . . . 6 |- ( ph -> F e. _V )
22		imasex 13603	. . . . . 6 |- ( ( F e. _V /\ R e. Z ) -> ( F "s R ) e. _V )
23	21, 10, 22	syl2anc 415	. . . . 5 |- ( ph -> ( F "s R ) e. _V )
24	2, 23	eqeltrd 2315	. . . 4 |- ( ph -> U e. _V )
25		mulridx 13462	. . . 4 |- .r = Slot ( .r ` ndx )
26		mulrslid 13463	. . . . 5 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
27	26	simpri 113	. . . 4 |- ( .r ` ndx ) e. NN
28	24, 25, 27	strndxid 13358	. . 3 |- ( ph -> ( U ` ( .r ` ndx ) ) = ( .r ` U ) )
29	27	a1i 9	. . . 4 |- ( ph -> ( .r ` ndx ) e. NN )
30		vex 2824	. . . . . . . . . . . 12 |- p e. _V
31		fvexg 5709	. . . . . . . . . . . 12 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
32	21, 30, 31	sylancl 417	. . . . . . . . . . 11 |- ( ph -> ( F ` p ) e. _V )
33		vex 2824	. . . . . . . . . . . 12 |- q e. _V
34		fvexg 5709	. . . . . . . . . . . 12 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
35	21, 33, 34	sylancl 417	. . . . . . . . . . 11 |- ( ph -> ( F ` q ) e. _V )
36		opexg 4363	. . . . . . . . . . 11 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
37	32, 35, 36	syl2anc 415	. . . . . . . . . 10 |- ( ph -> <. ( F ` p ) , ( F ` q ) >. e. _V )
38	26	slotex 13357	. . . . . . . . . . . . . 14 |- ( R e. Z -> ( .r ` R ) e. _V )
39	10, 38	syl 14	. . . . . . . . . . . . 13 |- ( ph -> ( .r ` R ) e. _V )
40	5, 39	eqeltrid 2325	. . . . . . . . . . . 12 |- ( ph -> .x. e. _V )
41	33	a1i 9	. . . . . . . . . . . 12 |- ( ph -> q e. _V )
42		ovexg 6109	. . . . . . . . . . . 12 |- ( ( p e. _V /\ .x. e. _V /\ q e. _V ) -> ( p .x. q ) e. _V )
43	30, 40, 41, 42	mp3an2i 1383	. . . . . . . . . . 11 |- ( ph -> ( p .x. q ) e. _V )
44		fvexg 5709	. . . . . . . . . . 11 |- ( ( F e. _V /\ ( p .x. q ) e. _V ) -> ( F ` ( p .x. q ) ) e. _V )
45	21, 43, 44	syl2anc 415	. . . . . . . . . 10 |- ( ph -> ( F ` ( p .x. q ) ) e. _V )
46		opexg 4363	. . . . . . . . . 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 415	. . . . . . . . 9 |- ( ph -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. e. _V )
48		snexg 4316	. . . . . . . . 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 2624	. . . . . . 7 |- ( ph -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
51		iunexg 6338	. . . . . . 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 415	. . . . . 6 |- ( ph -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
53	52	ralrimivw 2624	. . . . 5 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
54		iunexg 6338	. . . . 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 415	. . . 4 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } e. _V )
56		basendxnmulrndx 13465	. . . . 5 |- ( Base ` ndx ) =/= ( .r ` ndx )
57	56	a1i 9	. . . 4 |- ( ph -> ( Base ` ndx ) =/= ( .r ` ndx ) )
58		plusgndxnmulrndx 13464	. . . . 5 |- ( +g ` ndx ) =/= ( .r ` ndx )
59	58	a1i 9	. . . 4 |- ( ph -> ( +g ` ndx ) =/= ( .r ` ndx ) )
60		fvtp3g 5916	. . . 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 1279	. . 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 2280	. 2 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } = ( .r ` U ) )
63	1, 62	eqtr4id 2290	1 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .x. q ) ) >. } )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   _Vcvv 2821   {csn 3705   {ctp 3707   <.cop 3708   U_ciun 4007   Fn wfn 5367   -->wf 5368   -onto->wfo 5370   ` cfv 5372  (class class class)co 6075   NNcn 9283   ndxcnx 13327  Slot cslot 13329   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   .scvsca 13412   "_s cimas 13599

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mulr 13422  df-iimas 13601

This theorem is referenced by:  imasmulfn  13618  imasmulval  13619  imasmulf  13620  qusmulval  13635  qusmulf  13636