Theorem rng1zrlem 14198

Description: Lemma for rng1zr 14199 and srg1zr 14230. (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		pm4.24 395	. 2 |- ( B = { Z } <-> ( B = { Z } /\ B = { Z } ) )
2		simp1l 1048	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> R e. Mgm )
3		simp3 1026	. . . 4 |- ( ( ( R e. Mgm /\ (mulGrp ` R ) e. Mgm ) /\ ( .+ Fn ( B X. B ) /\ .* Fn ( B X. B ) ) /\ Z e. B ) -> Z e. B )
4		simp2l 1050	. . . 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 13631	. . . 4 |- ( ( R e. Mgm /\ Z e. B /\ .+ Fn ( B X. B ) ) -> ( B = { Z } <-> .+ = { <. <. Z , Z >. , Z >. } ) )
8	2, 3, 4, 7	syl3anc 1274	. . 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 2234	. . . . . . . 8 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 5	mgpbasg 14165	. . . . . . 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 1045	. . . . 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 2243	. . . 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 1049	. . . . 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 2304	. . . . . . 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 1043	. . . . 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 14163	. . . . . . . . . . . 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 5451	. . . . . . . . . 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 1044	. . . . . 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 4780	. . . . . . 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 5452	. . . . . 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 2234	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
30		eqid 2234	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
31	29, 30	mgmb1mgm1 13631	. . . . 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 1274	. . . 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 2240	. . . . . 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 2243	. . . 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 473	. 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 >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   {csn 3694   <.cop 3697   X. cxp 4752   Fn wfn 5352   ` cfv 5357   Basecbs 13296   +g cplusg 13374   .rcmulr 13375  Mgmcmgm 13617  mulGrpcmgp 14159

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-plusf 13618  df-mgm 13619  df-mgp 14160

This theorem is referenced by:  rng1zr  14199  srg1zr  14230