Theorem rnghmf1o 20432

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 20418	. . . . 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 20427	. . . . . . 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 19223	. . . . . . 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 2737	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 = 𝐹)
14		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
15	14, 7	mgpbas 20126	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
16	15	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅)))
17		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
18	17, 8	mgpbas 20126	. . . . . . . 8 ⊢ 𝐶 = (Base‘(mulGrp‘𝑆))
19	18	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆)))
20	13, 16, 19	f1oeq123d 6774	. . . . . 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 20423	. . . . . . 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 2736	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
25		eqid 2736	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆))
26	24, 25	mgmhmf1o 18668	. . . . . . 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 20422	. . 3 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ ((𝑆 ∈ Rng ∧ 𝑅 ∈ Rng) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))))
32	3, 30, 31	sylanbrc 584	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
33	7, 8	rnghmf 20428	. . . . 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 6669	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹 Fn 𝐵)
36	8, 7	rnghmf 20428	. . . . 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 6669	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹 Fn 𝐶)
39		dff1o4 6788	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
40	35, 38, 39	sylanbrc 584	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
41	32, 40	impbida 801	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ^◡ccnv 5630   Fn wfn 6493  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  Basecbs 17179   MgmHom cmgmhm 18658   GrpHom cghm 19187  mulGrpcmgp 20121  Rngcrng 20133   RngHom crnghm 20414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-plusg 17233  df-mgm 18608  df-mgmhm 18660  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-ghm 19188  df-abl 19758  df-mgp 20122  df-rng 20134  df-rnghm 20416

This theorem is referenced by:  isrngim2  20433