Theorem rnghmf1o 20530

Description: A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rnghmf1o.b	⊢ 𝐵 = (Base‘𝑅)
rnghmf1o.c	⊢ 𝐶 = (Base‘𝑆)

Assertion

Ref	Expression
rnghmf1o	⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))

Proof of Theorem rnghmf1o

Step	Hyp	Ref	Expression
1		rnghmrcl 20516	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2	1	ancomd 466	. . . 4 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
3	2	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
4		simpr 489	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶)
5		rnghmghm 20525	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
6	5	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
7		rnghmf1o.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
8		rnghmf1o.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑆)
9	7, 8	ghmf1o 19313	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅)))
10	9	bicomd 226	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶))
11	6, 10	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶))
12	4, 11	mpbird 260	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 GrpHom 𝑅))
13		eqidd 2764	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 = 𝐹)
14		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
15	14, 7	mgpbas 20216	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
16	15	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅)))
17		eqid 2763	. . . . . . . . 9 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
18	17, 8	mgpbas 20216	. . . . . . . 8 ⊢ 𝐶 = (Base‘(mulGrp‘𝑆))
19	18	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆)))
20	13, 16, 19	f1oeq123d 6814	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))))
21	20	biimpa 481	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)))
22	14, 17	rnghmmgmhm 20521	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
23	22	adantr 485	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
24		eqid 2763	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
25		eqid 2763	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆))
26	24, 25	mgmhmf1o 18753	. . . . . . 7 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)) → (𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆)) ↔ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅))))
27	26	bicomd 226	. . . . . 6 ⊢ (𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)) → (◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))))
28	23, 27	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)) ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))))
29	21, 28	mpbird 260	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))
30	12, 29	jca 520	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅))))
31	17, 14	isrnghmmul 20520	. . 3 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ ((𝑆 ∈ Rng ∧ 𝑅 ∈ Rng) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))))
32	3, 30, 31	sylanbrc 594	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
33	7, 8	rnghmf 20526	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)
34	33	adantr 485	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵⟶𝐶)
35	34	ffnd 6706	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹 Fn 𝐵)
36	8, 7	rnghmf 20526	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
37	36	adantl 486	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
38	37	ffnd 6706	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹 Fn 𝐶)
39		dff1o4 6829	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
40	35, 38, 39	sylanbrc 594	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
41	32, 40	impbida 812	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ^◡ccnv 5660   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  Basecbs 17264   MgmHom cmgmhm 18743   GrpHom cghm 19278  mulGrpcmgp 20211  Rngcrng 20225   RngHom crnghm 20512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-plusg 17318  df-mgm 18693  df-mgmhm 18745  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-ghm 19279  df-abl 19848  df-mgp 20212  df-rng 20226  df-rnghm 20514

This theorem is referenced by:  isrngim2  20531