Theorem frlmup2 20138

Description: The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)

Hypotheses

Ref	Expression
frlmup.f	|- F = ( R freeLMod I )
frlmup.b	|- B = ( Base ` F )
frlmup.c	|- C = ( Base ` T )
frlmup.v	|- .x. = ( .s ` T )
frlmup.e	|- E = ( x e. B |-> ( T gsumg ( x oF .x. A ) ) )
frlmup.t	|- ( ph -> T e. LMod )
frlmup.i	|- ( ph -> I e. X )
frlmup.r	|- ( ph -> R = (Scalar ` T ) )
frlmup.a	|- ( ph -> A : I --> C )
frlmup.y	|- ( ph -> Y e. I )
frlmup.u	|- U = ( R unitVec I )

Assertion

Ref	Expression
frlmup2	|- ( ph -> ( E ` ( U ` Y ) ) = ( A ` Y ) )

Distinct variable groups:   x, R   x, I   x, F   x, B   x, C   x, .x.   x, A   x, X   ph, x   x, Y   x, U   x, T

Allowed substitution hint:   E( x)

Proof of Theorem frlmup2

Step	Hyp	Ref	Expression
1		frlmup.r	. . . . . 6 |- ( ph -> R = (Scalar ` T ) )
2		frlmup.t	. . . . . . 7 |- ( ph -> T e. LMod )
3		eqid 2622	. . . . . . . 8 |- (Scalar ` T ) = (Scalar ` T )
4	3	lmodring 18871	. . . . . . 7 |- ( T e. LMod -> (Scalar ` T ) e. Ring )
5	2, 4	syl 17	. . . . . 6 |- ( ph -> (Scalar ` T ) e. Ring )
6	1, 5	eqeltrd 2701	. . . . 5 |- ( ph -> R e. Ring )
7		frlmup.i	. . . . 5 |- ( ph -> I e. X )
8		frlmup.u	. . . . . 6 |- U = ( R unitVec I )
9		frlmup.f	. . . . . 6 |- F = ( R freeLMod I )
10		frlmup.b	. . . . . 6 |- B = ( Base ` F )
11	8, 9, 10	uvcff 20130	. . . . 5 |- ( ( R e. Ring /\ I e. X ) -> U : I --> B )
12	6, 7, 11	syl2anc 693	. . . 4 |- ( ph -> U : I --> B )
13		frlmup.y	. . . 4 |- ( ph -> Y e. I )
14	12, 13	ffvelrnd 6360	. . 3 |- ( ph -> ( U ` Y ) e. B )
15		oveq1 6657	. . . . 5 |- ( x = ( U ` Y ) -> ( x oF .x. A ) = ( ( U ` Y ) oF .x. A ) )
16	15	oveq2d 6666	. . . 4 |- ( x = ( U ` Y ) -> ( T gsumg ( x oF .x. A ) ) = ( T gsumg ( ( U ` Y ) oF .x. A ) ) )
17		frlmup.e	. . . 4 |- E = ( x e. B |-> ( T gsumg ( x oF .x. A ) ) )
18		ovex 6678	. . . 4 |- ( T gsumg ( ( U ` Y ) oF .x. A ) ) e. _V
19	16, 17, 18	fvmpt 6282	. . 3 |- ( ( U ` Y ) e. B -> ( E ` ( U ` Y ) ) = ( T gsumg ( ( U ` Y ) oF .x. A ) ) )
20	14, 19	syl 17	. 2 |- ( ph -> ( E ` ( U ` Y ) ) = ( T gsumg ( ( U ` Y ) oF .x. A ) ) )
21		frlmup.c	. . 3 |- C = ( Base ` T )
22		eqid 2622	. . 3 |- ( 0g ` T ) = ( 0g ` T )
23		lmodcmn 18911	. . . 4 |- ( T e. LMod -> T e. CMnd )
24		cmnmnd 18208	. . . 4 |- ( T e. CMnd -> T e. Mnd )
25	2, 23, 24	3syl 18	. . 3 |- ( ph -> T e. Mnd )
26		eqid 2622	. . . 4 |- ( Base ` (Scalar ` T ) ) = ( Base ` (Scalar ` T ) )
27		frlmup.v	. . . 4 |- .x. = ( .s ` T )
28		eqid 2622	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
29	9, 28, 10	frlmbasf 20104	. . . . . 6 |- ( ( I e. X /\ ( U ` Y ) e. B ) -> ( U ` Y ) : I --> ( Base ` R ) )
30	7, 14, 29	syl2anc 693	. . . . 5 |- ( ph -> ( U ` Y ) : I --> ( Base ` R ) )
31	1	fveq2d 6195	. . . . . 6 |- ( ph -> ( Base ` R ) = ( Base ` (Scalar ` T ) ) )
32	31	feq3d 6032	. . . . 5 |- ( ph -> ( ( U ` Y ) : I --> ( Base ` R ) <-> ( U ` Y ) : I --> ( Base ` (Scalar ` T ) ) ) )
33	30, 32	mpbid 222	. . . 4 |- ( ph -> ( U ` Y ) : I --> ( Base ` (Scalar ` T ) ) )
34		frlmup.a	. . . 4 |- ( ph -> A : I --> C )
35	3, 26, 27, 21, 2, 33, 34, 7	lcomf 18902	. . 3 |- ( ph -> ( ( U ` Y ) oF .x. A ) : I --> C )
36		ffn 6045	. . . . . . . 8 |- ( ( U ` Y ) : I --> ( Base ` R ) -> ( U ` Y ) Fn I )
37	30, 36	syl 17	. . . . . . 7 |- ( ph -> ( U ` Y ) Fn I )
38	37	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( U ` Y ) Fn I )
39		ffn 6045	. . . . . . . 8 |- ( A : I --> C -> A Fn I )
40	34, 39	syl 17	. . . . . . 7 |- ( ph -> A Fn I )
41	40	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> A Fn I )
42	7	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> I e. X )
43		eldifi 3732	. . . . . . 7 |- ( x e. ( I \ { Y } ) -> x e. I )
44	43	adantl 482	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> x e. I )
45		fnfvof 6911	. . . . . 6 |- ( ( ( ( U ` Y ) Fn I /\ A Fn I ) /\ ( I e. X /\ x e. I ) ) -> ( ( ( U ` Y ) oF .x. A ) ` x ) = ( ( ( U ` Y ) ` x ) .x. ( A ` x ) ) )
46	38, 41, 42, 44, 45	syl22anc 1327	. . . . 5 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( ( U ` Y ) oF .x. A ) ` x ) = ( ( ( U ` Y ) ` x ) .x. ( A ` x ) ) )
47	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> R e. Ring )
48	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> Y e. I )
49		eldifsni 4320	. . . . . . . . . 10 |- ( x e. ( I \ { Y } ) -> x =/= Y )
50	49	necomd 2849	. . . . . . . . 9 |- ( x e. ( I \ { Y } ) -> Y =/= x )
51	50	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> Y =/= x )
52		eqid 2622	. . . . . . . 8 |- ( 0g ` R ) = ( 0g ` R )
53	8, 47, 42, 48, 44, 51, 52	uvcvv0 20129	. . . . . . 7 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( U ` Y ) ` x ) = ( 0g ` R ) )
54	1	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( 0g ` R ) = ( 0g ` (Scalar ` T ) ) )
55	54	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( 0g ` R ) = ( 0g ` (Scalar ` T ) ) )
56	53, 55	eqtrd 2656	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( U ` Y ) ` x ) = ( 0g ` (Scalar ` T ) ) )
57	56	oveq1d 6665	. . . . 5 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( ( U ` Y ) ` x ) .x. ( A ` x ) ) = ( ( 0g ` (Scalar ` T ) ) .x. ( A ` x ) ) )
58	2	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> T e. LMod )
59		ffvelrn 6357	. . . . . . 7 |- ( ( A : I --> C /\ x e. I ) -> ( A ` x ) e. C )
60	34, 43, 59	syl2an 494	. . . . . 6 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( A ` x ) e. C )
61		eqid 2622	. . . . . . 7 |- ( 0g ` (Scalar ` T ) ) = ( 0g ` (Scalar ` T ) )
62	21, 3, 27, 61, 22	lmod0vs 18896	. . . . . 6 |- ( ( T e. LMod /\ ( A ` x ) e. C ) -> ( ( 0g ` (Scalar ` T ) ) .x. ( A ` x ) ) = ( 0g ` T ) )
63	58, 60, 62	syl2anc 693	. . . . 5 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( 0g ` (Scalar ` T ) ) .x. ( A ` x ) ) = ( 0g ` T ) )
64	46, 57, 63	3eqtrd 2660	. . . 4 |- ( ( ph /\ x e. ( I \ { Y } ) ) -> ( ( ( U ` Y ) oF .x. A ) ` x ) = ( 0g ` T ) )
65	35, 64	suppss 7325	. . 3 |- ( ph -> ( ( ( U ` Y ) oF .x. A ) supp ( 0g ` T ) ) C_ { Y } )
66	21, 22, 25, 7, 13, 35, 65	gsumpt 18361	. 2 |- ( ph -> ( T gsumg ( ( U ` Y ) oF .x. A ) ) = ( ( ( U ` Y ) oF .x. A ) ` Y ) )
67		fnfvof 6911	. . . 4 |- ( ( ( ( U ` Y ) Fn I /\ A Fn I ) /\ ( I e. X /\ Y e. I ) ) -> ( ( ( U ` Y ) oF .x. A ) ` Y ) = ( ( ( U ` Y ) ` Y ) .x. ( A ` Y ) ) )
68	37, 40, 7, 13, 67	syl22anc 1327	. . 3 |- ( ph -> ( ( ( U ` Y ) oF .x. A ) ` Y ) = ( ( ( U ` Y ) ` Y ) .x. ( A ` Y ) ) )
69		eqid 2622	. . . . . 6 |- ( 1r ` R ) = ( 1r ` R )
70	8, 6, 7, 13, 69	uvcvv1 20128	. . . . 5 |- ( ph -> ( ( U ` Y ) ` Y ) = ( 1r ` R ) )
71	1	fveq2d 6195	. . . . 5 |- ( ph -> ( 1r ` R ) = ( 1r ` (Scalar ` T ) ) )
72	70, 71	eqtrd 2656	. . . 4 |- ( ph -> ( ( U ` Y ) ` Y ) = ( 1r ` (Scalar ` T ) ) )
73	72	oveq1d 6665	. . 3 |- ( ph -> ( ( ( U ` Y ) ` Y ) .x. ( A ` Y ) ) = ( ( 1r ` (Scalar ` T ) ) .x. ( A ` Y ) ) )
74	34, 13	ffvelrnd 6360	. . . 4 |- ( ph -> ( A ` Y ) e. C )
75		eqid 2622	. . . . 5 |- ( 1r ` (Scalar ` T ) ) = ( 1r ` (Scalar ` T ) )
76	21, 3, 27, 75	lmodvs1 18891	. . . 4 |- ( ( T e. LMod /\ ( A ` Y ) e. C ) -> ( ( 1r ` (Scalar ` T ) ) .x. ( A ` Y ) ) = ( A ` Y ) )
77	2, 74, 76	syl2anc 693	. . 3 |- ( ph -> ( ( 1r ` (Scalar ` T ) ) .x. ( A ` Y ) ) = ( A ` Y ) )
78	68, 73, 77	3eqtrd 2660	. 2 |- ( ph -> ( ( ( U ` Y ) oF .x. A ) ` Y ) = ( A ` Y ) )
79	20, 66, 78	3eqtrd 2660	1 |- ( ph -> ( E ` ( U ` Y ) ) = ( A ` Y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  CMndccmn 18193   1rcur 18501   Ringcrg 18547   LModclmod 18863   freeLMod cfrlm 20090   unitVec cuvc 20121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091  df-uvc 20122

This theorem is referenced by:  frlmup3  20139  frlmup4  20140