Theorem 2stropg 13452

Description: The other slot 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
2stropg	|- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )

Proof of Theorem 2stropg

Step	Hyp	Ref	Expression
1		2str.e	. . 3 |- E = Slot N
2		2str.n	. . 3 |- N e. NN
3	1, 2	ndxslid 13355	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
4		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
5		basendxnn 13386	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) e. NN )
7		simpl 109	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> B e. V )
8		opexg 4363	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
9	6, 7, 8	syl2anc 415	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. _V )
10	1, 2	ndxarg 13353	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 2	eqeltri 2311	. . . . . 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 4363	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
15	12, 13, 14	syl2anc 415	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
16		prexg 4344	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( E ` ndx ) , .+ >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
17	9, 15, 16	syl2anc 415	. . 3 |- ( ( B e. V /\ .+ e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
18	4, 17	eqeltrid 2325	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	5	nnrei 9292	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 13385	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4155	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 8412	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5426	. . . 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	6, 12, 7, 13, 24, 25	syl221anc 1289	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	4	funeqi 5393	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 134	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid2g 3812	. . . 4 |- ( <. ( E ` ndx ) , .+ >. e. _V -> <. ( E ` ndx ) , .+ >. e. { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
30	15, 29	syl 14	. . 3 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
31	30, 4	eleqtrrdi 2332	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. G )
32	3, 18, 28, 31	strslfvd 13372	1 |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   {cpr 3706   <.cop 3708   class class class wbr 4125   Fun wfun 5366   ` cfv 5372   1c1 8170   < clt 8350   NNcn 9283   ndxcnx 13327  Slot cslot 13329   Basecbs 13330

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-pre-ltirr 8281

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336

This theorem is referenced by:  grpplusgg  13459  eltpsg  15064