Theorem lsppropd 14064

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 2231	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4	3	pweqd 3611	. . 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 14063	. . . . 5 |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
14	13	rabeqdv 2757	. . . 4 |- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
15	14	inteqd 3880	. . 3 |- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } )
16	4, 15	mpteq12dv 4116	. 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 2196	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2196	. . . 4 |- ( LSubSp ` K ) = ( LSubSp ` K )
19		eqid 2196	. . . 4 |- ( LSpan ` K ) = ( LSpan ` K )
20	17, 18, 19	lspfval 14020	. . 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 2196	. . . 4 |- ( Base ` L ) = ( Base ` L )
23		eqid 2196	. . . 4 |- ( LSubSp ` L ) = ( LSubSp ` L )
24		eqid 2196	. . . 4 |- ( LSpan ` L ) = ( LSpan ` L )
25	22, 23, 24	lspfval 14020	. . 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 2239	1 |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   {crab 2479   C_ wss 3157   ~Pcpw 3606   |^|cint 3875   |-> cmpt 4095   ` cfv 5259  (class class class)co 5925   Basecbs 12703   +g cplusg 12780  Scalarcsca 12783   .scvsca 12784   LSubSpclss 13984   LSpanclspn 14018

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 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-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-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  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-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-lssm 13985  df-lsp 14019

This theorem is referenced by:  lidlrsppropdg  14127