Theorem lmhmf1o 18867

Description: A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypotheses

Ref	Expression
lmhmf1o.x	⊢ 𝑋 = (Base‘𝑆)
lmhmf1o.y	⊢ 𝑌 = (Base‘𝑇)

Assertion

Ref	Expression
lmhmf1o	⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

Proof of Theorem lmhmf1o

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmhmf1o.y	. . 3 ⊢ 𝑌 = (Base‘𝑇)
2		eqid 2610	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
3		eqid 2610	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
4		eqid 2610	. . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
5		eqid 2610	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
6		eqid 2610	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7		lmhmlmod2 18853	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
8	7	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑇 ∈ LMod)
9		lmhmlmod1 18854	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑆 ∈ LMod)
11	5, 4	lmhmsca 18851	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
12	11	eqcomd 2616	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
13	12	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14		lmghm 18852	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15		lmhmf1o.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
16	15, 1	ghmf1o 17513	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))
17	14, 16	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))
18	17	biimpa 500	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
19		simpll 786	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20	13	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
21	20	eleq2d 2673	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
22	21	biimpar 501	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
23	22	adantrr 749	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24		f1ocnv 6062	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6050	. . . . . . . . . 10 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
26	24, 25	syl 17	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌⟶𝑋)
27	26	adantl 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
28	27	ffvelrnda 6267	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑏 ∈ 𝑌) → (◡𝐹‘𝑏) ∈ 𝑋)
29	28	adantrl 748	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘𝑏) ∈ 𝑋)
30		eqid 2610	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31	5, 30, 15, 3, 2	lmhmlin 18856	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (◡𝐹‘𝑏) ∈ 𝑋) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘(◡𝐹‘𝑏))))
32	19, 23, 29, 31	syl3anc 1318	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘(◡𝐹‘𝑏))))
33		f1ocnvfv2 6433	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑏 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	33	ad2ant2l 778	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	34	oveq2d 6565	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)𝑏))
36	32, 35	eqtrd 2644	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)𝑏))
37		simplr 788	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹:𝑋–1-1-onto→𝑌)
38	10	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑆 ∈ LMod)
39	15, 5, 3, 30	lmodvscl 18703	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (◡𝐹‘𝑏) ∈ 𝑋) → (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋)
40	38, 23, 29, 39	syl3anc 1318	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋)
41		f1ocnvfv 6434	. . . . 5 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋) → ((𝐹‘(𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)𝑏) → (◡𝐹‘(𝑎( ·𝑠 ‘𝑇)𝑏)) = (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))))
42	37, 40, 41	syl2anc 691	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → ((𝐹‘(𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))) = (𝑎( ·𝑠 ‘𝑇)𝑏) → (◡𝐹‘(𝑎( ·𝑠 ‘𝑇)𝑏)) = (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏))))
43	36, 42	mpd 15	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘(𝑎( ·𝑠 ‘𝑇)𝑏)) = (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)))
44	1, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43	islmhmd 18860	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 LMHom 𝑆))
45	15, 1	lmhmf 18855	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋⟶𝑌)
46		ffn 5958	. . . . 5 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋)
47	45, 46	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
48	47	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
49	1, 15	lmhmf 18855	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 LMHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
50	49	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
51		ffn 5958	. . . 4 ⊢ (◡𝐹:𝑌⟶𝑋 → ◡𝐹 Fn 𝑌)
52	50, 51	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹 Fn 𝑌)
53		dff1o4 6058	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
54	48, 52, 53	sylanbrc 695	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
55	44, 54	impbida 873	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ^◡ccnv 5037   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·_𝑠 cvsca 15772   GrpHom cghm 17480  LModclmod 18686   LMHom clmhm 18840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ghm 17481  df-lmod 18688  df-lmhm 18843

This theorem is referenced by:  islmim2  18887