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Theorem lmhmf1o 20522
Description: A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
lmhmf1o.x 𝑋 = (Baseβ€˜π‘†)
lmhmf1o.y π‘Œ = (Baseβ€˜π‘‡)
Assertion
Ref Expression
lmhmf1o (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ ◑𝐹 ∈ (𝑇 LMHom 𝑆)))

Proof of Theorem lmhmf1o
Dummy variables π‘Ž 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmhmf1o.y . . 3 π‘Œ = (Baseβ€˜π‘‡)
2 eqid 2733 . . 3 ( ·𝑠 β€˜π‘‡) = ( ·𝑠 β€˜π‘‡)
3 eqid 2733 . . 3 ( ·𝑠 β€˜π‘†) = ( ·𝑠 β€˜π‘†)
4 eqid 2733 . . 3 (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘‡)
5 eqid 2733 . . 3 (Scalarβ€˜π‘†) = (Scalarβ€˜π‘†)
6 eqid 2733 . . 3 (Baseβ€˜(Scalarβ€˜π‘‡)) = (Baseβ€˜(Scalarβ€˜π‘‡))
7 lmhmlmod2 20508 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑇 ∈ LMod)
87adantr 482 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝑇 ∈ LMod)
9 lmhmlmod1 20509 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
109adantr 482 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝑆 ∈ LMod)
115, 4lmhmsca 20506 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘†))
1211eqcomd 2739 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (Scalarβ€˜π‘†) = (Scalarβ€˜π‘‡))
1312adantr 482 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (Scalarβ€˜π‘†) = (Scalarβ€˜π‘‡))
14 lmghm 20507 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Baseβ€˜π‘†)
1615, 1ghmf1o 19043 . . . . 5 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ ◑𝐹 ∈ (𝑇 GrpHom 𝑆)))
1714, 16syl 17 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ ◑𝐹 ∈ (𝑇 GrpHom 𝑆)))
1817biimpa 478 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 766 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))
2013fveq2d 6847 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (Baseβ€˜(Scalarβ€˜π‘†)) = (Baseβ€˜(Scalarβ€˜π‘‡)))
2120eleq2d 2820 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ↔ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡))))
2221biimpar 479 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡))) β†’ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)))
2322adantrr 716 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)))
24 f1ocnv 6797 . . . . . . . . . 10 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘Œβ€“1-1-onto→𝑋)
25 f1of 6785 . . . . . . . . . 10 (◑𝐹:π‘Œβ€“1-1-onto→𝑋 β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
2624, 25syl 17 . . . . . . . . 9 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
2726adantl 483 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
2827ffvelcdmda 7036 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑏 ∈ π‘Œ) β†’ (β—‘πΉβ€˜π‘) ∈ 𝑋)
2928adantrl 715 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (β—‘πΉβ€˜π‘) ∈ 𝑋)
30 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘†)) = (Baseβ€˜(Scalarβ€˜π‘†))
315, 30, 15, 3, 2lmhmlin 20511 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ∧ (β—‘πΉβ€˜π‘) ∈ 𝑋) β†’ (πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜(β—‘πΉβ€˜π‘))))
3219, 23, 29, 31syl3anc 1372 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜(β—‘πΉβ€˜π‘))))
33 f1ocnvfv2 7224 . . . . . . 7 ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ 𝑏 ∈ π‘Œ) β†’ (πΉβ€˜(β—‘πΉβ€˜π‘)) = 𝑏)
3433ad2ant2l 745 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (πΉβ€˜(β—‘πΉβ€˜π‘)) = 𝑏)
3534oveq2d 7374 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (π‘Ž( ·𝑠 β€˜π‘‡)(πΉβ€˜(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)𝑏))
3632, 35eqtrd 2773 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)𝑏))
37 simplr 768 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
3810adantr 482 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ 𝑆 ∈ LMod)
3915, 5, 3, 30lmodvscl 20354 . . . . . 6 ((𝑆 ∈ LMod ∧ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ∧ (β—‘πΉβ€˜π‘) ∈ 𝑋) β†’ (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘)) ∈ 𝑋)
4038, 23, 29, 39syl3anc 1372 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘)) ∈ 𝑋)
41 f1ocnvfv 7225 . . . . 5 ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘)) ∈ 𝑋) β†’ ((πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)𝑏) β†’ (β—‘πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘‡)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))))
4237, 40, 41syl2anc 585 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ ((πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))) = (π‘Ž( ·𝑠 β€˜π‘‡)𝑏) β†’ (β—‘πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘‡)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘))))
4336, 42mpd 15 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡)) ∧ 𝑏 ∈ π‘Œ)) β†’ (β—‘πΉβ€˜(π‘Ž( ·𝑠 β€˜π‘‡)𝑏)) = (π‘Ž( ·𝑠 β€˜π‘†)(β—‘πΉβ€˜π‘)))
441, 2, 3, 4, 5, 6, 8, 10, 13, 18, 43islmhmd 20515 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 20510 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4645ffnd 6670 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 Fn 𝑋)
4746adantr 482 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ 𝐹 Fn 𝑋)
481, 15lmhmf 20510 . . . . 5 (◑𝐹 ∈ (𝑇 LMHom 𝑆) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
4948adantl 483 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
5049ffnd 6670 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ ◑𝐹 Fn π‘Œ)
51 dff1o4 6793 . . 3 (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ (𝐹 Fn 𝑋 ∧ ◑𝐹 Fn π‘Œ))
5247, 50, 51sylanbrc 584 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
5344, 52impbida 800 1 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ ◑𝐹 ∈ (𝑇 LMHom 𝑆)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β—‘ccnv 5633   Fn wfn 6492  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Scalarcsca 17141   ·𝑠 cvsca 17142   GrpHom cghm 19010  LModclmod 20336   LMHom clmhm 20495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-ghm 19011  df-lmod 20338  df-lmhm 20498
This theorem is referenced by:  islmim2  20542
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