Theorem lsppropd 13931

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.) (Revised by AV, 24-Apr-2024.)

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 ) ) )
lsppropd.v1	|- ( ph -> K e. X )
lsppropd.v2	|- ( ph -> L e. Y )

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

Allowed substitution hints:   X( x, y)   Y( x, 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 2228	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4	3	pweqd 3607	. . 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		lsppropd.v1	. . . . . 6 |- ( ph -> K e. X )
12		lsppropd.v2	. . . . . 6 |- ( ph -> L e. Y )
13	1, 2, 5, 6, 7, 8, 9, 10, 11, 12	lsspropdg 13930	. . . . 5 |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
14	13	rabeqdv 2754	. . . 4 |- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
15	14	inteqd 3876	. . 3 |- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } )
16	4, 15	mpteq12dv 4112	. 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 } ) )
17		eqid 2193	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2193	. . . 4 |- ( LSubSp ` K ) = ( LSubSp ` K )
19		eqid 2193	. . . 4 |- ( LSpan ` K ) = ( LSpan ` K )
20	17, 18, 19	lspfval 13887	. . 3 |- ( K e. X -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) )
21	11, 20	syl 14	. 2 |- ( ph -> ( LSpan ` K ) = ( s e. ~P ( Base ` K ) |-> |^| { t e. ( LSubSp ` K ) | s C_ t } ) )
22		eqid 2193	. . . 4 |- ( Base ` L ) = ( Base ` L )
23		eqid 2193	. . . 4 |- ( LSubSp ` L ) = ( LSubSp ` L )
24		eqid 2193	. . . 4 |- ( LSpan ` L ) = ( LSpan ` L )
25	22, 23, 24	lspfval 13887	. . 3 |- ( L e. Y -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) )
26	12, 25	syl 14	. 2 |- ( ph -> ( LSpan ` L ) = ( s e. ~P ( Base ` L ) |-> |^| { t e. ( LSubSp ` L ) | s C_ t } ) )
27	16, 21, 26	3eqtr4d 2236	1 |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {crab 2476   C_ wss 3154   ~Pcpw 3602   |^|cint 3871   |-> cmpt 4091   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698  Scalarcsca 12701   .scvsca 12702   LSubSpclss 13851   LSpanclspn 13885

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-lssm 13852  df-lsp 13886

This theorem is referenced by:  lidlrsppropdg  13994