Theorem rng1zrlem 14233

Description: Lemma for rng1zr 14234 and srg1zr 14265. (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 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 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 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑅 ∈ Mgm)
3		simp3 1030	. . . 4 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵)
4		simp2l 1054	. . . 4 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → + Fn (𝐵 × 𝐵))
5		rng1zr.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
6		rng1zr.p	. . . . 5 ⊢ + = (+g‘𝑅)
7	5, 6	mgmb1mgm1 13665	. . . 4 ⊢ ((𝑅 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
8	2, 3, 4, 7	syl3anc 1278	. . 3 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
9		eqid 2238	. . . . . . . 8 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
10	9, 5	mgpbasg 14200	. . . . . . 7 ⊢ (𝑅 ∈ Mgm → 𝐵 = (Base‘(mulGrp‘𝑅)))
11	10	adantr 276	. . . . . 6 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → 𝐵 = (Base‘(mulGrp‘𝑅)))
12	11	3ad2ant1 1049	. . . . 5 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝐵 = (Base‘(mulGrp‘𝑅)))
13	12	eqeq1d 2247	. . . 4 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ (Base‘(mulGrp‘𝑅)) = {𝑍}))
14		simp1r 1053	. . . . 5 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (mulGrp‘𝑅) ∈ Mgm)
15	11	eleq2d 2308	. . . . . . 7 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → (𝑍 ∈ 𝐵 ↔ 𝑍 ∈ (Base‘(mulGrp‘𝑅))))
16	15	biimpa 296	. . . . . 6 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (Base‘(mulGrp‘𝑅)))
17	16	3adant2 1047	. . . . 5 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ (Base‘(mulGrp‘𝑅)))
18		rng1zr.t	. . . . . . . . . . . . 13 ⊢ ∗ = (.r‘𝑅)
19	9, 18	mgpplusgg 14198	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Mgm → ∗ = (+g‘(mulGrp‘𝑅)))
20	19	adantr 276	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → ∗ = (+g‘(mulGrp‘𝑅)))
21	20	fneq1d 5466	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → ( ∗ Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)))
22	21	biimpd 144	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → ( ∗ Fn (𝐵 × 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)))
23	22	adantld 278	. . . . . . . 8 ⊢ ((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) → (( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵)))
24	23	imp 124	. . . . . . 7 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵))) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
25	24	3adant3 1048	. . . . . 6 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵))
26	12	sqxpeqd 4795	. . . . . . 7 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 × 𝐵) = ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅))))
27	26	fneq2d 5467	. . . . . 6 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) Fn (𝐵 × 𝐵) ↔ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))))
28	25, 27	mpbid 147	. . . . 5 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (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 13665	. . . . 5 ⊢ (((mulGrp‘𝑅) ∈ Mgm ∧ 𝑍 ∈ (Base‘(mulGrp‘𝑅)) ∧ (+g‘(mulGrp‘𝑅)) Fn ((Base‘(mulGrp‘𝑅)) × (Base‘(mulGrp‘𝑅)))) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
32	14, 17, 28, 31	syl3anc 1278	. . . 4 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((Base‘(mulGrp‘𝑅)) = {𝑍} ↔ (+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
33	19	eqcomd 2244	. . . . . 6 ⊢ (𝑅 ∈ Mgm → (+g‘(mulGrp‘𝑅)) = ∗ )
34	2, 33	syl 14	. . . . 5 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (+g‘(mulGrp‘𝑅)) = ∗ )
35	34	eqeq1d 2247	. . . 4 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((+g‘(mulGrp‘𝑅)) = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ↔ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
36	13, 32, 35	3bitrd 214	. . 3 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
37	8, 36	anbi12d 477	. 2 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → ((𝐵 = {𝑍} ∧ 𝐵 = {𝑍}) ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
38	1, 37	bitrid 192	1 ⊢ (((𝑅 ∈ Mgm ∧ (mulGrp‘𝑅) ∈ Mgm) ∧ ( + Fn (𝐵 × 𝐵) ∧ ∗ Fn (𝐵 × 𝐵)) ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ ∗ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {csn 3705  ⟨cop 3708   × cxp 4767   Fn wfn 5367  ‘cfv 5372  Basecbs 13330  +_gcplusg 13408  ._rcmulr 13409  Mgmcmgm 13651  mulGrpcmgp 14194

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 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 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-plusf 13652  df-mgm 13653  df-mgp 14195

This theorem is referenced by:  rng1zr  14234  srg1zr  14265