Theorem rnghmf1o 20434

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 20420	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))
2	1	ancomd 460	. . . 4 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
3	2	adantr 479	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → (𝑆 ∈ Rng ∧ 𝑅 ∈ Rng))
4		simpr 483	. . . . 5 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹:𝐵–1-1-onto→𝐶)
5		rnghmghm 20429	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
6	5	adantr 479	. . . . . 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 19242	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 GrpHom 𝑅)))
10	9	bicomd 222	. . . . . 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 256	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 GrpHom 𝑅))
13		eqidd 2727	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 = 𝐹)
14		eqid 2726	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
15	14, 7	mgpbas 20123	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
16	15	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐵 = (Base‘(mulGrp‘𝑅)))
17		eqid 2726	. . . . . . . . 9 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
18	17, 8	mgpbas 20123	. . . . . . . 8 ⊢ 𝐶 = (Base‘(mulGrp‘𝑆))
19	18	a1i 11	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐶 = (Base‘(mulGrp‘𝑆)))
20	13, 16, 19	f1oeq123d 6837	. . . . . 6 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:(Base‘(mulGrp‘𝑅))–1-1-onto→(Base‘(mulGrp‘𝑆))))
21	20	biimpa 475	. . . . 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 20425	. . . . . . 7 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
23	22	adantr 479	. . . . . 6 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → 𝐹 ∈ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆)))
24		eqid 2726	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
25		eqid 2726	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑆)) = (Base‘(mulGrp‘𝑆))
26	24, 25	mgmhmf1o 18693	. . . . . . 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 222	. . . . . 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 256	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))
30	12, 29	jca 510	. . 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 20424	. . 3 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) ↔ ((𝑆 ∈ Rng ∧ 𝑅 ∈ Rng) ∧ (◡𝐹 ∈ (𝑆 GrpHom 𝑅) ∧ ◡𝐹 ∈ ((mulGrp‘𝑆) MgmHom (mulGrp‘𝑅)))))
32	3, 30, 31	sylanbrc 581	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝐵–1-1-onto→𝐶) → ◡𝐹 ∈ (𝑆 RngHom 𝑅))
33	7, 8	rnghmf 20430	. . . . 5 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → 𝐹:𝐵⟶𝐶)
34	33	adantr 479	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵⟶𝐶)
35	34	ffnd 6729	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹 Fn 𝐵)
36	8, 7	rnghmf 20430	. . . . 5 ⊢ (◡𝐹 ∈ (𝑆 RngHom 𝑅) → ◡𝐹:𝐶⟶𝐵)
37	36	adantl 480	. . . 4 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹:𝐶⟶𝐵)
38	37	ffnd 6729	. . 3 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → ◡𝐹 Fn 𝐶)
39		dff1o4 6851	. . 3 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐶))
40	35, 38, 39	sylanbrc 581	. 2 ⊢ ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ ◡𝐹 ∈ (𝑆 RngHom 𝑅)) → 𝐹:𝐵–1-1-onto→𝐶)
41	32, 40	impbida 799	1 ⊢ (𝐹 ∈ (𝑅 RngHom 𝑆) → (𝐹:𝐵–1-1-onto→𝐶 ↔ ◡𝐹 ∈ (𝑆 RngHom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ^◡ccnv 5681   Fn wfn 6549  ⟶wf 6550  –1-1-onto→wf1o 6553  ‘cfv 6554  (class class class)co 7424  Basecbs 17213   MgmHom cmgmhm 18683   GrpHom cghm 19206  mulGrpcmgp 20117  Rngcrng 20135   RngHom crnghm 20416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-plusg 17279  df-mgm 18633  df-mgmhm 18685  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-ghm 19207  df-abl 19781  df-mgp 20118  df-rng 20136  df-rnghm 20418

This theorem is referenced by:  isrngim2  20435