Theorem 2strbasg 12496

Description: The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)

Hypotheses

Ref	Expression
2str.g	|- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
2str.e	|- E = Slot N
2str.l	|- 1 < N
2str.n	|- N e. NN

Assertion

Ref	Expression
2strbasg	|- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )

Proof of Theorem 2strbasg

Step	Hyp	Ref	Expression
1		baseslid 12450	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
3		basendxnn 12449	. . . . . 6 |- ( Base ` ndx ) e. NN
4	3	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) e. NN )
5		simpl 108	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> B e. V )
6		opexg 4206	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
7	4, 5, 6	syl2anc 409	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. _V )
8		2str.e	. . . . . . . 8 |- E = Slot N
9		2str.n	. . . . . . . 8 |- N e. NN
10	8, 9	ndxarg 12417	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 9	eqeltri 2239	. . . . . 6 |- ( E ` ndx ) e. NN
12	11	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( E ` ndx ) e. NN )
13		simpr 109	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> .+ e. W )
14		opexg 4206	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
15	12, 13, 14	syl2anc 409	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
16		prexg 4189	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( E ` ndx ) , .+ >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
17	7, 15, 16	syl2anc 409	. . 3 |- ( ( B e. V /\ .+ e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
18	2, 17	eqeltrid 2253	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	3	nnrei 8866	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 12448	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4012	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 7995	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5238	. . . 4 |- ( ( ( ( Base ` ndx ) e. NN /\ ( E ` ndx ) e. NN ) /\ ( B e. V /\ .+ e. W ) /\ ( Base ` ndx ) =/= ( E ` ndx ) ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
26	4, 12, 5, 13, 24, 25	syl221anc 1239	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	2	funeqi 5209	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 133	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid1g 3680	. . . 4 |- ( <. ( Base ` ndx ) , B >. e. _V -> <. ( Base ` ndx ) , B >. e. { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
30	7, 29	syl 14	. . 3 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
31	30, 2	eleqtrrdi 2260	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. G )
32	1, 18, 28, 31	strslfvd 12435	1 |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336   _Vcvv 2726   {cpr 3577   <.cop 3579   class class class wbr 3982   Fun wfun 5182   ` cfv 5188   1c1 7754   < clt 7933   NNcn 8857   ndxcnx 12391  Slot cslot 12393   Basecbs 12394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850  ax-pre-ltirr 7865

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fv 5196  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-inn 8858  df-ndx 12397  df-slot 12398  df-base 12400

This theorem is referenced by:  grpbaseg  12503  eltpsg  12678