Theorem 2stropg 13334

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 13237	. 2 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
4		2str.g	. . 3 |- G = { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. }
5		basendxnn 13268	. . . . . 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 4344	. . . . 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 13235	. . . . . . 7 |- ( E ` ndx ) = N
11	10, 2	eqeltri 2305	. . . . . 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 4344	. . . . 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 4325	. . . 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 2319	. 2 |- ( ( B e. V /\ .+ e. W ) -> G e. _V )
19	5	nnrei 9246	. . . . . 6 |- ( Base ` ndx ) e. RR
20		2str.l	. . . . . . 7 |- 1 < N
21		basendx 13267	. . . . . . 7 |- ( Base ` ndx ) = 1
22	20, 21, 10	3brtr4i 4139	. . . . . 6 |- ( Base ` ndx ) < ( E ` ndx )
23	19, 22	ltneii 8370	. . . . 5 |- ( Base ` ndx ) =/= ( E ` ndx )
24	23	a1i 9	. . . 4 |- ( ( B e. V /\ .+ e. W ) -> ( Base ` ndx ) =/= ( E ` ndx ) )
25		funprg 5406	. . . 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 1285	. . 3 |- ( ( B e. V /\ .+ e. W ) -> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
27	4	funeqi 5373	. . 3 |- ( Fun G <-> Fun { <. ( Base ` ndx ) , B >. , <. ( E ` ndx ) , .+ >. } )
28	26, 27	sylibr 134	. 2 |- ( ( B e. V /\ .+ e. W ) -> Fun G )
29		prid2g 3796	. . . 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 2326	. 2 |- ( ( B e. V /\ .+ e. W ) -> <. ( E ` ndx ) , .+ >. e. G )
32	3, 18, 28, 31	strslfvd 13254	1 |- ( ( B e. V /\ .+ e. W ) -> .+ = ( E ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2813   {cpr 3690   <.cop 3692   class class class wbr 4109   Fun wfun 5346   ` cfv 5352   1c1 8128   < clt 8308   NNcn 9237   ndxcnx 13209  Slot cslot 13211   Basecbs 13212

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-pre-ltirr 8239

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218

This theorem is referenced by:  grpplusgg  13341  eltpsg  14905