Theorem rnghmf1o 20356

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 20342	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2	1	ancomd 461	. . . 4 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
3	2	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
4		simpr 484	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶)
5		rnghmghm 20351	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
6	5	adantr 480	. . . . . 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 19146	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅)))
10	9	bicomd 223	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ↔ 𝐹:𝐵–1-1-onto→𝐶))
11	6, 10	syl 17	. . . . 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 257	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 GrpHom 𝑅))
13		eqidd 2730	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 = 𝐹)
14		eqid 2729	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
15	14, 7	mgpbas 20049	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
16	15	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅)))
17		eqid 2729	. . . . . . . . 9 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
18	17, 8	mgpbas 20049	. . . . . . . 8 ⊢ 𝐶 = (Base‘(mulGrp‘𝑆))
19	18	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆)))
20	13, 16, 19	f1oeq123d 6762	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))))
21	20	biimpa 476	. . . . 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 20347	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
23	22	adantr 480	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
24		eqid 2729	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
25		eqid 2729	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆))
26	24, 25	mgmhmf1o 18593	. . . . . . 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 223	. . . . . 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 17	. . . . 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 257	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))
30	12, 29	jca 511	. . 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 20346	. . 3 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ ((𝑆 ∈ Rng ∧ 𝑅 ∈ Rng) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))))
32	3, 30, 31	sylanbrc 583	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
33	7, 8	rnghmf 20352	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)
34	33	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵⟶𝐶)
35	34	ffnd 6657	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹 Fn 𝐵)
36	8, 7	rnghmf 20352	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
37	36	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
38	37	ffnd 6657	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹 Fn 𝐶)
39		dff1o4 6776	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
40	35, 38, 39	sylanbrc 583	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
41	32, 40	impbida 800	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ^◡ccnv 5622   Fn wfn 6481  ⟶wf 6482  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353  Basecbs 17139   MgmHom cmgmhm 18583   GrpHom cghm 19110  mulGrpcmgp 20044  Rngcrng 20056   RngHom crnghm 20338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  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 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-map 8762  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17140  df-plusg 17193  df-mgm 18533  df-mgmhm 18585  df-sgrp 18612  df-mnd 18628  df-grp 18834  df-ghm 19111  df-abl 19681  df-mgp 20045  df-rng 20057  df-rnghm 20340

This theorem is referenced by:  isrngim2  20357