Theorem 2strbasg 12952

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 12889	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
3		basendxnn 12888	. . . . . 6 |- ( Base ` ndx ) e. NN
4	3	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) e. NN )
5		simpl 109	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> B e. V )
6		opexg 4272	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
7	4, 5, 6	syl2anc 411	. . . 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 12855	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 9	eqeltri 2278	. . . . . 6 |- ( E ` ndx ) e. NN
12	11	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( E ` ndx ) e. NN )
13		simpr 110	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> .+ e. W )
14		opexg 4272	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
15	12, 13, 14	syl2anc 411	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
16		prexg 4255	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( E ` ndx ) , .+ >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
17	7, 15, 16	syl2anc 411	. . 3 |- ( ( B e. V /\ .+ e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
18	2, 17	eqeltrid 2292	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	3	nnrei 9045	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 12887	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4074	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 8169	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5324	. . . 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 1261	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	2	funeqi 5292	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 134	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid1g 3737	. . . 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 2299	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. G )
32	1, 18, 28, 31	strslfvd 12874	1 |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   =/= wne 2376   _Vcvv 2772   {cpr 3634   <.cop 3636   class class class wbr 4044   Fun wfun 5265   ` cfv 5271   1c1 7926   < clt 8107   NNcn 9036   ndxcnx 12829  Slot cslot 12831   Basecbs 12832

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022  ax-pre-ltirr 8037

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fv 5279  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838

This theorem is referenced by:  grpbaseg  12959  eltpsg  14512