Theorem lsmpropd 18090

Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)

Hypotheses

Ref	Expression
lsmpropd.b1	|- ( ph -> B = ( Base ` K ) )
lsmpropd.b2	|- ( ph -> B = ( Base ` L ) )
lsmpropd.p	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
lsmpropd.v1	|- ( ph -> K e. _V )
lsmpropd.v2	|- ( ph -> L e. _V )

Assertion

Ref	Expression
lsmpropd	|- ( ph -> ( LSSum ` K ) = ( LSSum ` L ) )

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

Proof of Theorem lsmpropd

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

Step	Hyp	Ref	Expression
1		simp11 1091	. . . . . . 7 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> ph )
2		simp12 1092	. . . . . . . . 9 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> t e. ~P B )
3	2	elpwid 4170	. . . . . . . 8 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> t C_ B )
4		simp2 1062	. . . . . . . 8 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> x e. t )
5	3, 4	sseldd 3604	. . . . . . 7 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> x e. B )
6		simp13 1093	. . . . . . . . 9 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> u e. ~P B )
7	6	elpwid 4170	. . . . . . . 8 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> u C_ B )
8		simp3 1063	. . . . . . . 8 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> y e. u )
9	7, 8	sseldd 3604	. . . . . . 7 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> y e. B )
10		lsmpropd.p	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
11	1, 5, 9, 10	syl12anc 1324	. . . . . 6 |- ( ( ( ph /\ t e. ~P B /\ u e. ~P B ) /\ x e. t /\ y e. u ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
12	11	mpt2eq3dva 6719	. . . . 5 |- ( ( ph /\ t e. ~P B /\ u e. ~P B ) -> ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) = ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) )
13	12	rneqd 5353	. . . 4 |- ( ( ph /\ t e. ~P B /\ u e. ~P B ) -> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) = ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) )
14	13	mpt2eq3dva 6719	. . 3 |- ( ph -> ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
15		lsmpropd.b1	. . . . 5 |- ( ph -> B = ( Base ` K ) )
16	15	pweqd 4163	. . . 4 |- ( ph -> ~P B = ~P ( Base ` K ) )
17		mpt2eq12 6715	. . . 4 |- ( ( ~P B = ~P ( Base ` K ) /\ ~P B = ~P ( Base ` K ) ) -> ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) = ( t e. ~P ( Base ` K ) , u e. ~P ( Base ` K ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) )
18	16, 16, 17	syl2anc 693	. . 3 |- ( ph -> ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) = ( t e. ~P ( Base ` K ) , u e. ~P ( Base ` K ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) )
19		lsmpropd.b2	. . . . 5 |- ( ph -> B = ( Base ` L ) )
20	19	pweqd 4163	. . . 4 |- ( ph -> ~P B = ~P ( Base ` L ) )
21		mpt2eq12 6715	. . . 4 |- ( ( ~P B = ~P ( Base ` L ) /\ ~P B = ~P ( Base ` L ) ) -> ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) = ( t e. ~P ( Base ` L ) , u e. ~P ( Base ` L ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
22	20, 20, 21	syl2anc 693	. . 3 |- ( ph -> ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) = ( t e. ~P ( Base ` L ) , u e. ~P ( Base ` L ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
23	14, 18, 22	3eqtr3d 2664	. 2 |- ( ph -> ( t e. ~P ( Base ` K ) , u e. ~P ( Base ` K ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) = ( t e. ~P ( Base ` L ) , u e. ~P ( Base ` L ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
24		lsmpropd.v1	. . 3 |- ( ph -> K e. _V )
25		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
26		eqid 2622	. . . 4 |- ( +g ` K ) = ( +g ` K )
27		eqid 2622	. . . 4 |- ( LSSum ` K ) = ( LSSum ` K )
28	25, 26, 27	lsmfval 18053	. . 3 |- ( K e. _V -> ( LSSum ` K ) = ( t e. ~P ( Base ` K ) , u e. ~P ( Base ` K ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) )
29	24, 28	syl 17	. 2 |- ( ph -> ( LSSum ` K ) = ( t e. ~P ( Base ` K ) , u e. ~P ( Base ` K ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` K ) y ) ) ) )
30		lsmpropd.v2	. . 3 |- ( ph -> L e. _V )
31		eqid 2622	. . . 4 |- ( Base ` L ) = ( Base ` L )
32		eqid 2622	. . . 4 |- ( +g ` L ) = ( +g ` L )
33		eqid 2622	. . . 4 |- ( LSSum ` L ) = ( LSSum ` L )
34	31, 32, 33	lsmfval 18053	. . 3 |- ( L e. _V -> ( LSSum ` L ) = ( t e. ~P ( Base ` L ) , u e. ~P ( Base ` L ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
35	30, 34	syl 17	. 2 |- ( ph -> ( LSSum ` L ) = ( t e. ~P ( Base ` L ) , u e. ~P ( Base ` L ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` L ) y ) ) ) )
36	23, 29, 35	3eqtr4d 2666	1 |- ( ph -> ( LSSum ` K ) = ( LSSum ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   LSSumclsm 18049

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  ax-un 6949

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-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-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-lsm 18051

This theorem is referenced by:  hlhillsm  37248