Theorem lmhmf1o 21148

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 2763	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
3		eqid 2763	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
4		eqid 2763	. . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
5		eqid 2763	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
6		eqid 2763	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7		lmhmlmod2 21134	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
8	7	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑇 ∈ LMod)
9		lmhmlmod1 21135	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑆 ∈ LMod)
11	5, 4	lmhmsca 21132	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
12	11	eqcomd 2769	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
13	12	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14		lmghm 21133	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15		lmhmf1o.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
16	15, 1	ghmf1o 19319	. . . . 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 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
19		simpll 778	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20	13	fveq2d 6887	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
21	20	eleq2d 2849	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
22	21	biimpar 482	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
23	22	adantrr 729	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24		f1ocnv 6835	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6822	. . . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
28	27	ffvelcdmda 7081	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑏 ∈ 𝑌) → (◡𝐹‘𝑏) ∈ 𝑋)
29	28	adantrl 728	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘𝑏) ∈ 𝑋)
30		eqid 2763	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31	5, 30, 15, 3, 2	lmhmlin 21137	. . . . . 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 7277	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑏 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	33	ad2ant2l 758	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	34	oveq2d 7428	. . . . 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 2798	. . . 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 780	. . . . 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 485	. . . . . 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 20980	. . . . . 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 7278	. . . . 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 595	. . . 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 21141	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 LMHom 𝑆))
45	15, 1	lmhmf 21136	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋⟶𝑌)
46	45	ffnd 6708	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
47	46	adantr 485	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
48	1, 15	lmhmf 21136	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 LMHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
49	48	adantl 486	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
50	49	ffnd 6708	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹 Fn 𝑌)
51		dff1o4 6831	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
52	47, 50, 51	sylanbrc 594	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
53	44, 52	impbida 812	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ^◡ccnv 5662   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  Scalarcsca 17314   ·_𝑠 cvsca 17315   GrpHom cghm 19284  LModclmod 20962   LMHom clmhm 21121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  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 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-mgm 18699  df-sgrp 18778  df-mnd 18794  df-grp 19004  df-ghm 19285  df-lmod 20964  df-lmhm 21124

This theorem is referenced by:  islmim2  21168