Theorem lmhmf1o 20223

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 2738	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
3		eqid 2738	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
4		eqid 2738	. . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
5		eqid 2738	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
6		eqid 2738	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7		lmhmlmod2 20209	. . . 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 20210	. . . 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 20207	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
12	11	eqcomd 2744	. . . 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 20208	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15		lmhmf1o.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
16	15, 1	ghmf1o 18779	. . . . 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 476	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
19		simpll 763	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20	13	fveq2d 6760	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
21	20	eleq2d 2824	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
22	21	biimpar 477	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
23	22	adantrr 713	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24		f1ocnv 6712	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6700	. . . . . . . . . 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 6943	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑏 ∈ 𝑌) → (◡𝐹‘𝑏) ∈ 𝑋)
29	28	adantrl 712	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘𝑏) ∈ 𝑋)
30		eqid 2738	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31	5, 30, 15, 3, 2	lmhmlin 20212	. . . . . 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 1369	. . . . 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 7130	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑏 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	33	ad2ant2l 742	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	34	oveq2d 7271	. . . . 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 2778	. . . 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 765	. . . . 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 20055	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (◡𝐹‘𝑏) ∈ 𝑋) → (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋)
40	38, 23, 29, 39	syl3anc 1369	. . . . 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 7131	. . . . 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 583	. . . 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 20216	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 LMHom 𝑆))
45	15, 1	lmhmf 20211	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋⟶𝑌)
46	45	ffnd 6585	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
47	46	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
48	1, 15	lmhmf 20211	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 LMHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
49	48	adantl 481	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
50	49	ffnd 6585	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹 Fn 𝑌)
51		dff1o4 6708	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
52	47, 50, 51	sylanbrc 582	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
53	44, 52	impbida 797	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ^◡ccnv 5579   Fn wfn 6413  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Scalarcsca 16891   ·_𝑠 cvsca 16892   GrpHom cghm 18746  LModclmod 20038   LMHom clmhm 20196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-ghm 18747  df-lmod 20040  df-lmhm 20199

This theorem is referenced by:  islmim2  20243