Theorem imasival 13008

Description: Value of an image structure. The is a lemma for the theorems imasbas 13009, imasplusg 13010, and imasmulr 13011 and should not be needed once they are proved. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Jim Kingdon, 11-Mar-2025.) (New usage is discouraged.)

Hypotheses

Ref	Expression
imasval.u	|- ( ph -> U = ( F "s R ) )
imasval.v	|- ( ph -> V = ( Base ` R ) )
imasval.p	|- .+ = ( +g ` R )
imasval.m	|- .X. = ( .r ` R )
imasval.q	|- .x. = ( .s ` R )
imasval.a	|- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
imasval.t	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
imasval.f	|- ( ph -> F : V -onto-> B )
imasval.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
imasival	|- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )

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

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

Proof of Theorem imasival

Dummy variables f r v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasval.u	. 2 |- ( ph -> U = ( F "s R ) )
2		df-iimas 13004	. . . 4 |- "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +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 ( .r ` r ) q ) ) >. } >. } )
3	2	a1i 9	. . 3 |- ( ph -> "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +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 ( .r ` r ) q ) ) >. } >. } ) )
4		basfn 12761	. . . . . 6 |- Base Fn _V
5		vex 2766	. . . . . 6 |- r e. _V
6		funfvex 5578	. . . . . . 7 |- ( ( Fun Base /\ r e. dom Base ) -> ( Base ` r ) e. _V )
7	6	funfni 5361	. . . . . 6 |- ( ( Base Fn _V /\ r e. _V ) -> ( Base ` r ) e. _V )
8	4, 5, 7	mp2an 426	. . . . 5 |- ( Base ` r ) e. _V
9	8	a1i 9	. . . 4 |- ( ( ph /\ ( f = F /\ r = R ) ) -> ( Base ` r ) e. _V )
10		simplrl 535	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> f = F )
11	10	rneqd 4896	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran f = ran F )
12		imasval.f	. . . . . . . . 9 |- ( ph -> F : V -onto-> B )
13		forn 5486	. . . . . . . . 9 |- ( F : V -onto-> B -> ran F = B )
14	12, 13	syl 14	. . . . . . . 8 |- ( ph -> ran F = B )
15	14	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran F = B )
16	11, 15	eqtrd 2229	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran f = B )
17	16	opeq2d 3816	. . . . 5 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( Base ` ndx ) , ran f >. = <. ( Base ` ndx ) , B >. )
18		simplrr 536	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> r = R )
19	18	fveq2d 5565	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( Base ` r ) = ( Base ` R ) )
20		simpr 110	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> v = ( Base ` r ) )
21		imasval.v	. . . . . . . . . 10 |- ( ph -> V = ( Base ` R ) )
22	21	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> V = ( Base ` R ) )
23	19, 20, 22	3eqtr4d 2239	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> v = V )
24	10	fveq1d 5563	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` p ) = ( F ` p ) )
25	10	fveq1d 5563	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` q ) = ( F ` q ) )
26	24, 25	opeq12d 3817	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( f ` p ) , ( f ` q ) >. = <. ( F ` p ) , ( F ` q ) >. )
27	18	fveq2d 5565	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( +g ` r ) = ( +g ` R ) )
28		imasval.p	. . . . . . . . . . . . . 14 |- .+ = ( +g ` R )
29	27, 28	eqtr4di 2247	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( +g ` r ) = .+ )
30	29	oveqd 5942	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( +g ` r ) q ) = ( p .+ q ) )
31	10, 30	fveq12d 5568	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( p ( +g ` r ) q ) ) = ( F ` ( p .+ q ) ) )
32	26, 31	opeq12d 3817	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. )
33	32	sneqd 3636	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
34	23, 33	iuneq12d 3941	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
35	23, 34	iuneq12d 3941	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> 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 .+ q ) ) >. } )
36		imasval.a	. . . . . . . 8 |- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
37	36	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
38	35, 37	eqtr4d 2232	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = .+b )
39	38	opeq2d 3816	. . . . 5 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. = <. ( +g ` ndx ) , .+b >. )
40	18	fveq2d 5565	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .r ` r ) = ( .r ` R ) )
41		imasval.m	. . . . . . . . . . . . . 14 |- .X. = ( .r ` R )
42	40, 41	eqtr4di 2247	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .r ` r ) = .X. )
43	42	oveqd 5942	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( .r ` r ) q ) = ( p .X. q ) )
44	10, 43	fveq12d 5568	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( p ( .r ` r ) q ) ) = ( F ` ( p .X. q ) ) )
45	26, 44	opeq12d 3817	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. )
46	45	sneqd 3636	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
47	23, 46	iuneq12d 3941	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
48	23, 47	iuneq12d 3941	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
49		imasval.t	. . . . . . . 8 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
50	49	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
51	48, 50	eqtr4d 2232	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = .xb )
52	51	opeq2d 3816	. . . . 5 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. = <. ( .r ` ndx ) , .xb >. )
53	17, 39, 52	tpeq123d 3715	. . . 4 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. ( Base ` ndx ) , ran f >. , <. ( +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 ( .r ` r ) q ) ) >. } >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
54	9, 53	csbied 3131	. . 3 |- ( ( ph /\ ( f = F /\ r = R ) ) -> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +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 ( .r ` r ) q ) ) >. } >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
55		fof 5483	. . . . 5 |- ( F : V -onto-> B -> F : V --> B )
56	12, 55	syl 14	. . . 4 |- ( ph -> F : V --> B )
57		imasval.r	. . . . . . 7 |- ( ph -> R e. Z )
58	57	elexd 2776	. . . . . 6 |- ( ph -> R e. _V )
59		funfvex 5578	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
60	59	funfni 5361	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
61	4, 58, 60	sylancr 414	. . . . 5 |- ( ph -> ( Base ` R ) e. _V )
62	21, 61	eqeltrd 2273	. . . 4 |- ( ph -> V e. _V )
63	56, 62	fexd 5795	. . 3 |- ( ph -> F e. _V )
64		basendxnn 12759	. . . . 5 |- ( Base ` ndx ) e. NN
65		focdmex 6181	. . . . . 6 |- ( V e. _V -> ( F : V -onto-> B -> B e. _V ) )
66	62, 12, 65	sylc 62	. . . . 5 |- ( ph -> B e. _V )
67		opexg 4262	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. _V ) -> <. ( Base ` ndx ) , B >. e. _V )
68	64, 66, 67	sylancr 414	. . . 4 |- ( ph -> <. ( Base ` ndx ) , B >. e. _V )
69		plusgndxnn 12814	. . . . 5 |- ( +g ` ndx ) e. NN
70	63	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> F e. _V )
71		vex 2766	. . . . . . . . . . . . . . 15 |- p e. _V
72	71	a1i 9	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> p e. _V )
73		fvexg 5580	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ p e. _V ) -> ( F ` p ) e. _V )
74	70, 72, 73	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( F ` p ) e. _V )
75		vex 2766	. . . . . . . . . . . . . . 15 |- q e. _V
76	75	a1i 9	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> q e. _V )
77		fvexg 5580	. . . . . . . . . . . . . 14 |- ( ( F e. _V /\ q e. _V ) -> ( F ` q ) e. _V )
78	70, 76, 77	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( F ` q ) e. _V )
79		opexg 4262	. . . . . . . . . . . . 13 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
80	74, 78, 79	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
81		plusgslid 12815	. . . . . . . . . . . . . . . . . 18 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
82	81	slotex 12730	. . . . . . . . . . . . . . . . 17 |- ( R e. Z -> ( +g ` R ) e. _V )
83	57, 82	syl 14	. . . . . . . . . . . . . . . 16 |- ( ph -> ( +g ` R ) e. _V )
84	28, 83	eqeltrid 2283	. . . . . . . . . . . . . . 15 |- ( ph -> .+ e. _V )
85	84	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> .+ e. _V )
86		ovexg 5959	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ .+ e. _V /\ q e. _V ) -> ( p .+ q ) e. _V )
87	72, 85, 76, 86	syl3anc 1249	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( p .+ q ) e. _V )
88		fvexg 5580	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p .+ q ) e. _V ) -> ( F ` ( p .+ q ) ) e. _V )
89	70, 87, 88	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( F ` ( p .+ q ) ) e. _V )
90		opexg 4262	. . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p .+ q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. e. _V )
91	80, 89, 90	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. e. _V )
92		snexg 4218	. . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
93	91, 92	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
94	93	ralrimiva 2570	. . . . . . . . 9 |- ( ( ph /\ p e. V ) -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
95		iunexg 6185	. . . . . . . . 9 |- ( ( V e. _V /\ A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
96	62, 94, 95	syl2an2r 595	. . . . . . . 8 |- ( ( ph /\ p e. V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
97	96	ralrimiva 2570	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
98		iunexg 6185	. . . . . . 7 |- ( ( V e. _V /\ A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V ) -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
99	62, 97, 98	syl2anc 411	. . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } e. _V )
100	36, 99	eqeltrd 2273	. . . . 5 |- ( ph -> .+b e. _V )
101		opexg 4262	. . . . 5 |- ( ( ( +g ` ndx ) e. NN /\ .+b e. _V ) -> <. ( +g ` ndx ) , .+b >. e. _V )
102	69, 100, 101	sylancr 414	. . . 4 |- ( ph -> <. ( +g ` ndx ) , .+b >. e. _V )
103		mulrslid 12834	. . . . . 6 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
104	103	simpri 113	. . . . 5 |- ( .r ` ndx ) e. NN
105	103	slotex 12730	. . . . . . . . . . . . . . . . 17 |- ( R e. Z -> ( .r ` R ) e. _V )
106	57, 105	syl 14	. . . . . . . . . . . . . . . 16 |- ( ph -> ( .r ` R ) e. _V )
107	41, 106	eqeltrid 2283	. . . . . . . . . . . . . . 15 |- ( ph -> .X. e. _V )
108	107	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> .X. e. _V )
109		ovexg 5959	. . . . . . . . . . . . . 14 |- ( ( p e. _V /\ .X. e. _V /\ q e. _V ) -> ( p .X. q ) e. _V )
110	72, 108, 76, 109	syl3anc 1249	. . . . . . . . . . . . 13 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( p .X. q ) e. _V )
111		fvexg 5580	. . . . . . . . . . . . 13 |- ( ( F e. _V /\ ( p .X. q ) e. _V ) -> ( F ` ( p .X. q ) ) e. _V )
112	70, 110, 111	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> ( F ` ( p .X. q ) ) e. _V )
113		opexg 4262	. . . . . . . . . . . 12 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p .X. q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. e. _V )
114	80, 112, 113	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. e. _V )
115		snexg 4218	. . . . . . . . . . 11 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
116	114, 115	syl 14	. . . . . . . . . 10 |- ( ( ( ph /\ p e. V ) /\ q e. V ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
117	116	ralrimiva 2570	. . . . . . . . 9 |- ( ( ph /\ p e. V ) -> A. q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
118		iunexg 6185	. . . . . . . . 9 |- ( ( 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 )
119	62, 117, 118	syl2an2r 595	. . . . . . . 8 |- ( ( ph /\ p e. V ) -> U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
120	119	ralrimiva 2570	. . . . . . 7 |- ( ph -> A. p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
121		iunexg 6185	. . . . . . 7 |- ( ( 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 )
122	62, 120, 121	syl2anc 411	. . . . . 6 |- ( ph -> U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } e. _V )
123	49, 122	eqeltrd 2273	. . . . 5 |- ( ph -> .xb e. _V )
124		opexg 4262	. . . . 5 |- ( ( ( .r ` ndx ) e. NN /\ .xb e. _V ) -> <. ( .r ` ndx ) , .xb >. e. _V )
125	104, 123, 124	sylancr 414	. . . 4 |- ( ph -> <. ( .r ` ndx ) , .xb >. e. _V )
126		tpexg 4480	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( +g ` ndx ) , .+b >. e. _V /\ <. ( .r ` ndx ) , .xb >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } e. _V )
127	68, 102, 125, 126	syl3anc 1249	. . 3 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } e. _V )
128	3, 54, 63, 58, 127	ovmpod 6054	. 2 |- ( ph -> ( F "s R ) = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
129	1, 128	eqtrd 2229	1 |- ( ph -> U = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   [_csb 3084   {csn 3623   {ctp 3625   <.cop 3626   U_ciun 3917   ran crn 4665   Fn wfn 5254   -->wf 5255   -onto->wfo 5257   ` cfv 5259  (class class class)co 5925   e. cmpo 5927   NNcn 9007   ndxcnx 12700  Slot cslot 12702   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   .scvsca 12784   "_s cimas 13001

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-iimas 13004

This theorem is referenced by:  imasbas  13009  imasplusg  13010  imasmulr  13011