Theorem lmhmf1o 19820

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 2823	. . 3 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
3		eqid 2823	. . 3 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
4		eqid 2823	. . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
5		eqid 2823	. . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
6		eqid 2823	. . 3 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
7		lmhmlmod2 19806	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
8	7	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑇 ∈ LMod)
9		lmhmlmod1 19807	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
10	9	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → 𝑆 ∈ LMod)
11	5, 4	lmhmsca 19804	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
12	11	eqcomd 2829	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑆) = (Scalar‘𝑇))
13	12	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Scalar‘𝑆) = (Scalar‘𝑇))
14		lmghm 19805	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
15		lmhmf1o.x	. . . . . 6 ⊢ 𝑋 = (Base‘𝑆)
16	15, 1	ghmf1o 18390	. . . . 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 479	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 GrpHom 𝑆))
19		simpll 765	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝐹 ∈ (𝑆 LMHom 𝑇))
20	13	fveq2d 6676	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑇)))
21	20	eleq2d 2900	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑆)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑇))))
22	21	biimpar 480	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑇))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
23	22	adantrr 715	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
24		f1ocnv 6629	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6617	. . . . . . . . . 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 484	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹:𝑌⟶𝑋)
28	27	ffvelrnda 6853	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ 𝑏 ∈ 𝑌) → (◡𝐹‘𝑏) ∈ 𝑋)
29	28	adantrl 714	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (◡𝐹‘𝑏) ∈ 𝑋)
30		eqid 2823	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
31	5, 30, 15, 3, 2	lmhmlin 19809	. . . . . 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 1367	. . . . 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 7036	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑏 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	33	ad2ant2l 744	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ 𝑏 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	34	oveq2d 7174	. . . . 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 2858	. . . 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 767	. . . . 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 483	. . . . . 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 19653	. . . . . 6 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ (◡𝐹‘𝑏) ∈ 𝑋) → (𝑎( ·𝑠 ‘𝑆)(◡𝐹‘𝑏)) ∈ 𝑋)
40	38, 23, 29, 39	syl3anc 1367	. . . . 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 7037	. . . . 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 586	. . . 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 19813	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-onto→𝑌) → ◡𝐹 ∈ (𝑇 LMHom 𝑆))
45	15, 1	lmhmf 19808	. . . . 5 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝑋⟶𝑌)
46	45	ffnd 6517	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 Fn 𝑋)
47	46	adantr 483	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹 Fn 𝑋)
48	1, 15	lmhmf 19808	. . . . 5 ⊢ (◡𝐹 ∈ (𝑇 LMHom 𝑆) → ◡𝐹:𝑌⟶𝑋)
49	48	adantl 484	. . . 4 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹:𝑌⟶𝑋)
50	49	ffnd 6517	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → ◡𝐹 Fn 𝑌)
51		dff1o4 6625	. . 3 ⊢ (𝐹:𝑋–1-1-onto→𝑌 ↔ (𝐹 Fn 𝑋 ∧ ◡𝐹 Fn 𝑌))
52	47, 50, 51	sylanbrc 585	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◡𝐹 ∈ (𝑇 LMHom 𝑆)) → 𝐹:𝑋–1-1-onto→𝑌)
53	44, 52	impbida 799	1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋–1-1-onto→𝑌 ↔ ◡𝐹 ∈ (𝑇 LMHom 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ^◡ccnv 5556   Fn wfn 6352  ⟶wf 6353  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571   GrpHom cghm 18357  LModclmod 19636   LMHom clmhm 19793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-ghm 18358  df-lmod 19638  df-lmhm 19796

This theorem is referenced by:  islmim2  19840