Theorem ressascl 15022

Description: The lifting of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)

Hypotheses

Ref	Expression
ressascl.a	|- A = (algSc ` W )
ressascl.x	|- X = ( W ↾s S )

Assertion

Ref	Expression
ressascl	|- ( S e. (SubRing ` W ) -> A = (algSc ` X ) )

Proof of Theorem ressascl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrgrcl 14517	. . . . 5 |- ( S e. (SubRing ` W ) -> W e. Ring )
2		ressascl.x	. . . . . 6 |- X = ( W ↾s S )
3		eqid 2238	. . . . . 6 |- (Scalar ` W ) = (Scalar ` W )
4	2, 3	ressscag 13520	. . . . 5 |- ( ( W e. Ring /\ S e. (SubRing ` W ) ) -> (Scalar ` W ) = (Scalar ` X ) )
5	1, 4	mpancom 426	. . . 4 |- ( S e. (SubRing ` W ) -> (Scalar ` W ) = (Scalar ` X ) )
6	5	fveq2d 5697	. . 3 |- ( S e. (SubRing ` W ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` X ) ) )
7		eqid 2238	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
8	2, 7	ressvscag 13521	. . . . 5 |- ( ( W e. Ring /\ S e. (SubRing ` W ) ) -> ( .s ` W ) = ( .s ` X ) )
9	1, 8	mpancom 426	. . . 4 |- ( S e. (SubRing ` W ) -> ( .s ` W ) = ( .s ` X ) )
10		eqidd 2239	. . . 4 |- ( S e. (SubRing ` W ) -> x = x )
11		eqid 2238	. . . . 5 |- ( 1r ` W ) = ( 1r ` W )
12	2, 11	subrg1 14522	. . . 4 |- ( S e. (SubRing ` W ) -> ( 1r ` W ) = ( 1r ` X ) )
13	9, 10, 12	oveq123d 6100	. . 3 |- ( S e. (SubRing ` W ) -> ( x ( .s ` W ) ( 1r ` W ) ) = ( x ( .s ` X ) ( 1r ` X ) ) )
14	6, 13	mpteq12dv 4211	. 2 |- ( S e. (SubRing ` W ) -> ( x e. ( Base ` (Scalar ` W ) ) |-> ( x ( .s ` W ) ( 1r ` W ) ) ) = ( x e. ( Base ` (Scalar ` X ) ) |-> ( x ( .s ` X ) ( 1r ` X ) ) ) )
15		ressascl.a	. . 3 |- A = (algSc ` W )
16		eqid 2238	. . 3 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
17	15, 3, 16, 7, 11	asclfval 15004	. 2 |- A = ( x e. ( Base ` (Scalar ` W ) ) |-> ( x ( .s ` W ) ( 1r ` W ) ) )
18		eqid 2238	. . 3 |- (algSc ` X ) = (algSc ` X )
19		eqid 2238	. . 3 |- (Scalar ` X ) = (Scalar ` X )
20		eqid 2238	. . 3 |- ( Base ` (Scalar ` X ) ) = ( Base ` (Scalar ` X ) )
21		eqid 2238	. . 3 |- ( .s ` X ) = ( .s ` X )
22		eqid 2238	. . 3 |- ( 1r ` X ) = ( 1r ` X )
23	18, 19, 20, 21, 22	asclfval 15004	. 2 |- (algSc ` X ) = ( x e. ( Base ` (Scalar ` X ) ) |-> ( x ( .s ` X ) ( 1r ` X ) ) )
24	14, 17, 23	3eqtr4g 2296	1 |- ( S e. (SubRing ` W ) -> A = (algSc ` X ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   |-> cmpt 4190   ` cfv 5375  (class class class)co 6079   Basecbs 13335   ↾_s cress 13336  Scalarcsca 13417   .scvsca 13418   1rcur 14245   Ringcrg 14283  SubRingcsubrg 14508  algSccascl 14981

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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  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-rmo 2536  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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-subg 13956  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-ascl 14984

This theorem is referenced by: (None)