Theorem 2strbasg 13266

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 13203	. 2 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
2		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
3		basendxnn 13201	. . . . . 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 4326	. . . . 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 13168	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 9	eqeltri 2304	. . . . . 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 4326	. . . . 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 4307	. . . 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 2318	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	3	nnrei 9194	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 13200	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4123	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 8318	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5387	. . . 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 1285	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	2	funeqi 5354	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 134	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid1g 3779	. . . 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 2325	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. G )
32	1, 18, 28, 31	strslfvd 13187	1 |- ( ( B e. V /\ .+ e. W ) -> B = ( Base ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   {cpr 3674   <.cop 3676   class class class wbr 4093   Fun wfun 5327   ` cfv 5333   1c1 8076   < clt 8256   NNcn 9185   ndxcnx 13142  Slot cslot 13144   Basecbs 13145

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172  ax-pre-ltirr 8187

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151

This theorem is referenced by:  grpbaseg  13273  eltpsg  14834