Theorem lsppropd 19018

Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
lsspropd.b1	|- ( ph -> B = ( Base ` K ) )
lsspropd.b2	|- ( ph -> B = ( Base ` L ) )
lsspropd.w	|- ( ph -> B C_ W )
lsspropd.p	|- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
lsspropd.s1	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W )
lsspropd.s2	|- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
lsspropd.p1	|- ( ph -> P = ( Base ` (Scalar ` K ) ) )
lsspropd.p2	|- ( ph -> P = ( Base ` (Scalar ` L ) ) )
lsspropd.v1	|- ( ph -> K e. _V )
lsspropd.v2	|- ( ph -> L e. _V )

Assertion

Ref	Expression
lsppropd	|- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Distinct variable groups:   x, y, B   x, K, y   ph, x, y   x, W, y   x, L, y   x, P, y

Proof of Theorem lsppropd

Dummy variables s t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsspropd.b1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
2		lsspropd.b2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
3	1, 2	eqtr3d 2658	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4	3	pweqd 4163	. . 3 |- ( ph -> ~P ( Base ` K ) = ~P ( Base ` L ) )
5		lsspropd.w	. . . . . 6 |- ( ph -> B C_ W )
6		lsspropd.p	. . . . . 6 |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
7		lsspropd.s1	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) e. W )
8		lsspropd.s2	. . . . . 6 |- ( ( ph /\ ( x e. P /\ y e. B ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )
9		lsspropd.p1	. . . . . 6 |- ( ph -> P = ( Base ` (Scalar ` K ) ) )
10		lsspropd.p2	. . . . . 6 |- ( ph -> P = ( Base ` (Scalar ` L ) ) )
11	1, 2, 5, 6, 7, 8, 9, 10	lsspropd 19017	. . . . 5 |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
12		rabeq 3192	. . . . 5 |- ( ( LSubSp ` K ) = ( LSubSp ` L ) -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
13	11, 12	syl 17	. . . 4 |- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
14	13	inteqd 4480	. . 3 |- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } )
15	4, 14	mpteq12dv 4733	. 2 |- ( ph -> ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) )
16		lsspropd.v1	. . 3 |- ( ph -> K e. _V )
17		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2622	. . . 4 |- ( LSubSp ` K ) = ( LSubSp ` K )
19		eqid 2622	. . . 4 |- ( LSpan ` K ) = ( LSpan ` K )
20	17, 18, 19	lspfval 18973	. . 3 |- ( K e. _V -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) )
21	16, 20	syl 17	. 2 |- ( ph -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) )
22		lsspropd.v2	. . 3 |- ( ph -> L e. _V )
23		eqid 2622	. . . 4 |- ( Base ` L ) = ( Base ` L )
24		eqid 2622	. . . 4 |- ( LSubSp ` L ) = ( LSubSp ` L )
25		eqid 2622	. . . 4 |- ( LSpan ` L ) = ( LSpan ` L )
26	23, 24, 25	lspfval 18973	. . 3 |- ( L e. _V -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) )
27	22, 26	syl 17	. 2 |- ( ph -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) )
28	15, 21, 27	3eqtr4d 2666	1 |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   LSubSpclss 18932   LSpanclspn 18971

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lss 18933  df-lsp 18972

This theorem is referenced by:  lbspropd  19099  lidlrsppropd  19230