Theorem lmhmf1o 21196

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 2765	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
3		eqid 2765	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
4		eqid 2765	. . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
5		eqid 2765	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
6		eqid 2765	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7		lmhmlmod2 21182	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
8	7	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑇 ∈ LMod)
9		lmhmlmod1 21183	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑆 ∈ LMod)
11	5, 4	lmhmsca 21180	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
12	11	eqcomd 2771	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
13	12	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14		lmghm 21181	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15		lmhmf1o.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
16	15, 1	ghmf1o 19341	. . . . 5 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))
17	14, 16	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 GrpHom 𝑆)))
18	17	biimpa 482	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
19		simpll 779	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20	13	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
21	20	eleq2d 2851	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
22	21	biimpar 483	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
23	22	adantrr 730	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24		f1ocnv 6837	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6824	. . . . . . . . . 10 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
26	24, 25	syl 18	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌⟶𝑋)
27	26	adantl 487	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
28	27	ffvelcdmda 7083	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑏 ∈ 𝑌) → (◡𝐹‘𝑏) ∈ 𝑋)
29	28	adantrl 729	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘𝑏) ∈ 𝑋)
30		eqid 2765	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31	5, 30, 15, 3, 2	lmhmlin 21185	. . . . . 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 1398	. . . . 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 7281	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑏 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	33	ad2ant2l 759	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	34	oveq2d 7432	. . . . 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 2800	. . . 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 781	. . . . 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 486	. . . . . 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 21028	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (◡𝐹‘𝑏) ∈ 𝑋) → (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋)
40	38, 23, 29, 39	syl3anc 1398	. . . . 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 7282	. . . . 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 596	. . . 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 16	. . 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 21189	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 LMHom 𝑆))
45	15, 1	lmhmf 21184	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋⟶𝑌)
46	45	ffnd 6710	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
47	46	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
48	1, 15	lmhmf 21184	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 LMHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
49	48	adantl 487	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
50	49	ffnd 6710	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹 Fn 𝑌)
51		dff1o4 6833	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
52	47, 50, 51	sylanbrc 595	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
53	44, 52	impbida 813	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ^◡ccnv 5662   Fn wfn 6535  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7416  Basecbs 17286  Scalarcsca 17330   ·_𝑠 cvsca 17331   GrpHom cghm 19306  LModclmod 21010   LMHom clmhm 21169

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-map 8828  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-grp 19026  df-ghm 19307  df-lmod 21012  df-lmhm 21172

This theorem is used by:  islmim2  21216