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Theorem lmhmf1o 20656
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 2732 . . 3 ( ·𝑠 β€˜π‘‡) = ( ·𝑠 β€˜π‘‡)
3 eqid 2732 . . 3 ( ·𝑠 β€˜π‘†) = ( ·𝑠 β€˜π‘†)
4 eqid 2732 . . 3 (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘‡)
5 eqid 2732 . . 3 (Scalarβ€˜π‘†) = (Scalarβ€˜π‘†)
6 eqid 2732 . . 3 (Baseβ€˜(Scalarβ€˜π‘‡)) = (Baseβ€˜(Scalarβ€˜π‘‡))
7 lmhmlmod2 20642 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑇 ∈ LMod)
87adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝑇 ∈ LMod)
9 lmhmlmod1 20643 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
109adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝑆 ∈ LMod)
115, 4lmhmsca 20640 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (Scalarβ€˜π‘‡) = (Scalarβ€˜π‘†))
1211eqcomd 2738 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ (Scalarβ€˜π‘†) = (Scalarβ€˜π‘‡))
1312adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (Scalarβ€˜π‘†) = (Scalarβ€˜π‘‡))
14 lmghm 20641 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
15 lmhmf1o.x . . . . . 6 𝑋 = (Baseβ€˜π‘†)
1615, 1ghmf1o 19121 . . . . 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 477 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹 ∈ (𝑇 GrpHom 𝑆))
19 simpll 765 . . . . . 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 6895 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (Baseβ€˜(Scalarβ€˜π‘†)) = (Baseβ€˜(Scalarβ€˜π‘‡)))
2120eleq2d 2819 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)) ↔ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡))))
2221biimpar 478 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘‡))) β†’ π‘Ž ∈ (Baseβ€˜(Scalarβ€˜π‘†)))
2322adantrr 715 . . . . . 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 6845 . . . . . . . . . 10 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ ◑𝐹:π‘Œβ€“1-1-onto→𝑋)
25 f1of 6833 . . . . . . . . . 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 482 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
2827ffvelcdmda 7086 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑏 ∈ π‘Œ) β†’ (β—‘πΉβ€˜π‘) ∈ 𝑋)
2928adantrl 714 . . . . . 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 2732 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘†)) = (Baseβ€˜(Scalarβ€˜π‘†))
315, 30, 15, 3, 2lmhmlin 20645 . . . . . 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 1371 . . . . 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 7274 . . . . . . 7 ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ 𝑏 ∈ π‘Œ) β†’ (πΉβ€˜(β—‘πΉβ€˜π‘)) = 𝑏)
3433ad2ant2l 744 . . . . . 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 7424 . . . . 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 2772 . . . 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 767 . . . . 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 481 . . . . . 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 20488 . . . . . 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 1371 . . . . 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 7275 . . . . 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 584 . . . 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 20649 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ ◑𝐹 ∈ (𝑇 LMHom 𝑆))
4515, 1lmhmf 20644 . . . . 5 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
4645ffnd 6718 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 Fn 𝑋)
4746adantr 481 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ 𝐹 Fn 𝑋)
481, 15lmhmf 20644 . . . . 5 (◑𝐹 ∈ (𝑇 LMHom 𝑆) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
4948adantl 482 . . . 4 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ ◑𝐹:π‘ŒβŸΆπ‘‹)
5049ffnd 6718 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ ◑𝐹 Fn π‘Œ)
51 dff1o4 6841 . . 3 (𝐹:𝑋–1-1-ontoβ†’π‘Œ ↔ (𝐹 Fn 𝑋 ∧ ◑𝐹 Fn π‘Œ))
5247, 50, 51sylanbrc 583 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ ◑𝐹 ∈ (𝑇 LMHom 𝑆)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
5344, 52impbida 799 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 396   = wceq 1541   ∈ wcel 2106  β—‘ccnv 5675   Fn wfn 6538  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Scalarcsca 17199   ·𝑠 cvsca 17200   GrpHom cghm 19088  LModclmod 20470   LMHom clmhm 20629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-ghm 19089  df-lmod 20472  df-lmhm 20632
This theorem is referenced by:  islmim2  20676
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