Theorem lsppropd 14390

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 2264	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4	3	pweqd 3654	. . 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 14389	. . . . 5 |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
14	13	rabeqdv 2793	. . . 4 |- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
15	14	inteqd 3927	. . 3 |- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } )
16	4, 15	mpteq12dv 4165	. 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 2229	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2229	. . . 4 |- ( LSubSp ` K ) = ( LSubSp ` K )
19		eqid 2229	. . . 4 |- ( LSpan ` K ) = ( LSpan ` K )
20	17, 18, 19	lspfval 14346	. . 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 2229	. . . 4 |- ( Base ` L ) = ( Base ` L )
23		eqid 2229	. . . 4 |- ( LSubSp ` L ) = ( LSubSp ` L )
24		eqid 2229	. . . 4 |- ( LSpan ` L ) = ( LSpan ` L )
25	22, 23, 24	lspfval 14346	. . 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 2272	1 |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   ~Pcpw 3649   |^|cint 3922   |-> cmpt 4144   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105  Scalarcsca 13108   .scvsca 13109   LSubSpclss 14310   LSpanclspn 14344

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-lssm 14311  df-lsp 14345

This theorem is referenced by:  lidlrsppropdg  14453