Theorem 2stropg 12741

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 12646	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
4		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
5		basendxnn 12677	. . . . . 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 4258	. . . . 5 |- ( ( ( Base ` ndx ) e. NN /\ B e. V ) -> <. ( Base ` ndx ) , B >. e. _V )
9	6, 7, 8	syl2anc 411	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> <. ( Base ` ndx ) , B >. e. _V )
10	1, 2	ndxarg 12644	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 2	eqeltri 2266	. . . . . 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 4258	. . . . 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 4241	. . . 4 |- ( ( <. ( Base ` ndx ) , B >. e. _V /\ <. ( E ` ndx ) , .+ >. e. _V ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
17	9, 15, 16	syl2anc 411	. . 3 |- ( ( B e. V /\ .+ e. W ) -> { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } e. _V )
18	4, 17	eqeltrid 2280	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	5	nnrei 8993	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 12676	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4060	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 8118	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5305	. . . 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 1260	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	4	funeqi 5276	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 134	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid2g 3724	. . . 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 2287	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. G )
32	3, 18, 28, 31	strslfvd 12663	1 |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   {cpr 3620   <.cop 3622   class class class wbr 4030   Fun wfun 5249   ` cfv 5255   1c1 7875   < clt 8056   NNcn 8984   ndxcnx 12618  Slot cslot 12620   Basecbs 12621

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971  ax-pre-ltirr 7986

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627

This theorem is referenced by:  grpplusgg  12748  eltpsg  14219