Theorem rng1zrlem 14258

Description: Lemma for rng1zr 14259 and srg1zr 14291. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 18-Jun-2026.)

Hypotheses

Ref	Expression
rng1zr.b	|- B = ( Base ` R )
rng1zr.p	|- .+ = ( +g ` R )
rng1zr.t	|- .* = ( .r ` R )

Assertion

Ref	Expression
rng1zrlem	|- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

Proof of Theorem rng1zrlem

Step	Hyp	Ref	Expression
1		pm4.24 399	. 2 |- ( B = { Z } <-> ( B = { Z } /\ B = { Z } ) )
2		simp1l 1052	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mgm )
3		simp3 1030	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
4		simp2l 1054	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> .+ Fn ( B X. B ) )
5		rng1zr.b	. . . . 5 |- B = ( Base ` R )
6		rng1zr.p	. . . . 5 |- .+ = ( +g ` R )
7	5, 6	mgmb1mgm1 13688	. . . 4 |- ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
8	2, 3, 4, 7	syl3anc 1278	. . 3 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
9		eqid 2238	. . . . . . . 8 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 5	mgpbasg 14224	. . . . . . 7 |- ( R e. Mgm -> B = ( Base ` (mulGrp ` R ) ) )
11	10	adantr 276	. . . . . 6 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> B = ( Base ` (mulGrp ` R ) ) )
12	11	3ad2ant1 1049	. . . . 5 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
13	12	eqeq1d 2247	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( Base ` (mulGrp ` R ) ) = { Z } ) )
14		simp1r 1053	. . . . 5 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> (mulGrp ` R ) e. Mgm )
15	11	eleq2d 2308	. . . . . . 7 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> ( Z e. B <-> Z e. ( Base ` (mulGrp ` R ) ) ) )
16	15	biimpa 296	. . . . . 6 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ Z e. B ) -> Z e. ( Base ` (mulGrp ` R ) ) )
17	16	3adant2 1047	. . . . 5 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. ( Base ` (mulGrp ` R ) ) )
18		rng1zr.t	. . . . . . . . . . . . 13 |- .* = ( .r ` R )
19	9, 18	mgpplusgg 14221	. . . . . . . . . . . 12 |- ( R e. Mgm -> .* = ( +g ` (mulGrp ` R ) ) )
20	19	adantr 276	. . . . . . . . . . 11 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> .* = ( +g ` (mulGrp ` R ) ) )
21	20	fneq1d 5471	. . . . . . . . . 10 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> ( .* Fn ( B X. B ) <-> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) ) )
22	21	biimpd 144	. . . . . . . . 9 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> ( .* Fn ( B X. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) ) )
23	22	adantld 278	. . . . . . . 8 |- ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) -> ( ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) ) )
24	23	imp 124	. . . . . . 7 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
25	24	3adant3 1048	. . . . . 6 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) )
26	12	sqxpeqd 4800	. . . . . . 7 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B X. B ) = ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) )
27	26	fneq2d 5472	. . . . . 6 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( +g ` (mulGrp ` R ) ) Fn ( B X. B ) <-> ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) ) )
28	25, 27	mpbid 147	. . . . 5 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) )
29		eqid 2238	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
30		eqid 2238	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
31	29, 30	mgmb1mgm1 13688	. . . . 5 |- ( ( (mulGrp ` R ) e. Mgm /\ Z e. ( Base ` (mulGrp ` R ) ) /\ ( +g ` (mulGrp ` R ) ) Fn ( ( Base ` (mulGrp ` R ) ) X. ( Base ` (mulGrp ` R ) ) ) ) -> ( ( Base ` (mulGrp ` R ) ) = { Z } <-> ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
32	14, 17, 28, 31	syl3anc 1278	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( Base ` (mulGrp ` R ) ) = { Z } <-> ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } ) )
33	19	eqcomd 2244	. . . . . 6 |- ( R e. Mgm -> ( +g ` (mulGrp ` R ) ) = .* )
34	2, 33	syl 14	. . . . 5 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( +g ` (mulGrp ` R ) ) = .* )
35	34	eqeq1d 2247	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( +g ` (mulGrp ` R ) ) = { <. <. Z , Z >. , Z >. } <-> .* = { <. <. Z , Z >. , Z >. } ) )
36	13, 32, 35	3bitrd 214	. . 3 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> .* = { <. <. Z , Z >. , Z >. } ) )
37	8, 36	anbi12d 477	. 2 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( ( B = { Z } /\ B = { Z } ) <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )
38	1, 37	bitrid 192	1 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> ( B = { Z } <-> ( .+ = { <. <. Z , Z >. , Z >. } /\ .* = { <. <. Z , Z >. , Z >. } ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {csn 3709   <.cop 3712   X. cxp 4772   Fn wfn 5372   ` cfv 5377   Basecbs 13352   +g cplusg 13431   .rcmulr 13432  Mgmcmgm 13674  mulGrpcmgp 14217

This proof depends on 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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

This proof 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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-ltxr 8365  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-plusf 13675  df-mgm 13676  df-mgp 14218

This theorem is used by:  rng1zr  14259  srg1zr  14291