Theorem lsppropd 14269

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 2241	. . . 4 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
4	3	pweqd 3626	. . 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 14268	. . . . 5 |- ( ph -> ( LSubSp ` K ) = ( LSubSp ` L ) )
14	13	rabeqdv 2767	. . . 4 |- ( ph -> { t e. ( LSubSp ` K ) | s C_ t } = { t e. ( LSubSp ` L ) | s C_ t } )
15	14	inteqd 3896	. . 3 |- ( ph -> |^| { t e. ( LSubSp ` K ) | s C_ t } = |^| { t e. ( LSubSp ` L ) | s C_ t } )
16	4, 15	mpteq12dv 4134	. 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 2206	. . . 4 |- ( Base ` K ) = ( Base ` K )
18		eqid 2206	. . . 4 |- ( LSubSp ` K ) = ( LSubSp ` K )
19		eqid 2206	. . . 4 |- ( LSpan ` K ) = ( LSpan ` K )
20	17, 18, 19	lspfval 14225	. . 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 2206	. . . 4 |- ( Base ` L ) = ( Base ` L )
23		eqid 2206	. . . 4 |- ( LSubSp ` L ) = ( LSubSp ` L )
24		eqid 2206	. . . 4 |- ( LSpan ` L ) = ( LSpan ` L )
25	22, 23, 24	lspfval 14225	. . 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 2249	1 |- ( ph -> ( LSpan ` K ) = ( LSpan ` L ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   {crab 2489   C_ wss 3170   ~Pcpw 3621   |^|cint 3891   |-> cmpt 4113   ` cfv 5280  (class class class)co 5957   Basecbs 12907   +g cplusg 12984  Scalarcsca 12987   .scvsca 12988   LSubSpclss 14189   LSpanclspn 14223

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-cnex 8036  ax-resscn 8037  ax-1re 8039  ax-addrcl 8042

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-inn 9057  df-ndx 12910  df-slot 12911  df-base 12913  df-lssm 14190  df-lsp 14224

This theorem is referenced by:  lidlrsppropdg  14332