Theorem 2stropg 12061

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 11984	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
4		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
5		basendxnn 12014	. . . . . 6 |- ( Base ` ndx ) e. NN
6	5	a1i 9	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) e. NN )
7		simpl 108	. . . . 5 |- ( ( B e. V /\ .+ e. W ) -> B e. V )
8		opexg 4150	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
9	6, 7, 8	syl2anc 408	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. _V )
10	1, 2	ndxarg 11982	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 2	eqeltri 2212	. . . . . 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 4150	. . . . 5 |- ( ( ( E ` ndx ) e. NN /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
15	12, 13, 14	syl2anc 408	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. _V )
16		prexg 4133	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( E ` ndx ) , .+ >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
17	9, 15, 16	syl2anc 408	. . 3 |- ( ( B e. V /\ .+ e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
18	4, 17	eqeltrid 2226	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	5	nnrei 8729	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 12013	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 3958	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 7860	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5173	. . . 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 1227	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	4	funeqi 5144	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 133	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid2g 3628	. . . 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 2233	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. G )
32	3, 18, 28, 31	strslfvd 12000	1 |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2308   _Vcvv 2686   {cpr 3528   <.cop 3530   class class class wbr 3929   Fun wfun 5117   ` cfv 5123   1c1 7621   < clt 7800   NNcn 8720   ndxcnx 11956  Slot cslot 11958   Basecbs 11959

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-pre-ltirr 7732

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-pnf 7802  df-mnf 7803  df-ltxr 7805  df-inn 8721  df-ndx 11962  df-slot 11963  df-base 11965

This theorem is referenced by:  grpplusgg  12068  eltpsg  12207