MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pj1lmhm Structured version   Visualization version   GIF version
Theorem pj1lmhm 20704
Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1lmhm.l 𝐿 = (LSubSpβ€˜π‘Š)
pj1lmhm.s βŠ• = (LSSumβ€˜π‘Š)
pj1lmhm.z 0 = (0gβ€˜π‘Š)
pj1lmhm.p 𝑃 = (proj1β€˜π‘Š)
pj1lmhm.1 (πœ‘ β†’ π‘Š ∈ LMod)
pj1lmhm.2 (πœ‘ β†’ 𝑇 ∈ 𝐿)
pj1lmhm.3 (πœ‘ β†’ π‘ˆ ∈ 𝐿)
pj1lmhm.4 (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })
Assertion
Ref Expression
pj1lmhm (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) LMHom π‘Š))
Proof of Theorem pj1lmhm
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
2 pj1lmhm.s . . 3 βŠ• = (LSSumβ€˜π‘Š)
3 pj1lmhm.z . . 3 0 = (0gβ€˜π‘Š)
4 eqid 2733 . . 3 (Cntzβ€˜π‘Š) = (Cntzβ€˜π‘Š)
5 pj1lmhm.1 . . . . 5 (πœ‘ β†’ π‘Š ∈ LMod)
6 pj1lmhm.l . . . . . 6 𝐿 = (LSubSpβ€˜π‘Š)
76lsssssubg 20562 . . . . 5 (π‘Š ∈ LMod β†’ 𝐿 βŠ† (SubGrpβ€˜π‘Š))
85, 7syl 17 . . . 4 (πœ‘ β†’ 𝐿 βŠ† (SubGrpβ€˜π‘Š))
9 pj1lmhm.2 . . . 4 (πœ‘ β†’ 𝑇 ∈ 𝐿)
108, 9sseldd 3983 . . 3 (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
11 pj1lmhm.3 . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝐿)
128, 11sseldd 3983 . . 3 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
13 pj1lmhm.4 . . 3 (πœ‘ β†’ (𝑇 ∩ π‘ˆ) = { 0 })
14 lmodabl 20512 . . . . 5 (π‘Š ∈ LMod β†’ π‘Š ∈ Abel)
155, 14syl 17 . . . 4 (πœ‘ β†’ π‘Š ∈ Abel)
164, 15, 10, 12ablcntzd 19720 . . 3 (πœ‘ β†’ 𝑇 βŠ† ((Cntzβ€˜π‘Š)β€˜π‘ˆ))
17 pj1lmhm.p . . 3 𝑃 = (proj1β€˜π‘Š)
181, 2, 3, 4, 10, 12, 13, 16, 17pj1ghm 19566 . 2 (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) GrpHom π‘Š))
19 eqid 2733 . . 3 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
2019a1i 11 . 2 (πœ‘ β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š))
211, 2, 3, 4, 10, 12, 13, 16, 17pj1id 19562 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ)) β†’ 𝑦 = (((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)(+gβ€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦)))
2221adantrl 715 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑦 = (((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)(+gβ€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦)))
2322oveq2d 7422 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)𝑦) = (π‘₯( ·𝑠 β€˜π‘Š)(((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)(+gβ€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))))
245adantr 482 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ π‘Š ∈ LMod)
25 simprl 770 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
269adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑇 ∈ 𝐿)
27 eqid 2733 . . . . . . . . . . 11 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
2827, 6lssss 20540 . . . . . . . . . 10 (𝑇 ∈ 𝐿 β†’ 𝑇 βŠ† (Baseβ€˜π‘Š))
2926, 28syl 17 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑇 βŠ† (Baseβ€˜π‘Š))
3010adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
3112adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
3213adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (𝑇 ∩ π‘ˆ) = { 0 })
3316adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑇 βŠ† ((Cntzβ€˜π‘Š)β€˜π‘ˆ))
341, 2, 3, 4, 30, 31, 32, 33, 17pj1f 19560 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘‡π‘ƒπ‘ˆ):(𝑇 βŠ• π‘ˆ)βŸΆπ‘‡)
35 simprr 772 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))
3634, 35ffvelcdmd 7085 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦) ∈ 𝑇)
3729, 36sseldd 3983 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦) ∈ (Baseβ€˜π‘Š))
3811adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ π‘ˆ ∈ 𝐿)
3927, 6lssss 20540 . . . . . . . . . 10 (π‘ˆ ∈ 𝐿 β†’ π‘ˆ βŠ† (Baseβ€˜π‘Š))
4038, 39syl 17 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ π‘ˆ βŠ† (Baseβ€˜π‘Š))
411, 2, 3, 4, 30, 31, 32, 33, 17pj2f 19561 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘ˆπ‘ƒπ‘‡):(𝑇 βŠ• π‘ˆ)βŸΆπ‘ˆ)
4241, 35ffvelcdmd 7085 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦) ∈ π‘ˆ)
4340, 42sseldd 3983 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦) ∈ (Baseβ€˜π‘Š))
44 eqid 2733 . . . . . . . . 9 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
45 eqid 2733 . . . . . . . . 9 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
4627, 1, 19, 44, 45lmodvsdi 20488 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦) ∈ (Baseβ€˜π‘Š) ∧ ((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦) ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)(((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)(+gβ€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))) = ((π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))(+gβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))))
4724, 25, 37, 43, 46syl13anc 1373 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)(((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)(+gβ€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))) = ((π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))(+gβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))))
4823, 47eqtrd 2773 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)𝑦) = ((π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))(+gβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))))
496, 2lsmcl 20687 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝐿 ∧ π‘ˆ ∈ 𝐿) β†’ (𝑇 βŠ• π‘ˆ) ∈ 𝐿)
505, 9, 11, 49syl3anc 1372 . . . . . . . . 9 (πœ‘ β†’ (𝑇 βŠ• π‘ˆ) ∈ 𝐿)
5150adantr 482 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (𝑇 βŠ• π‘ˆ) ∈ 𝐿)
5219, 44, 45, 6lssvscl 20559 . . . . . . . 8 (((π‘Š ∈ LMod ∧ (𝑇 βŠ• π‘ˆ) ∈ 𝐿) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ (𝑇 βŠ• π‘ˆ))
5324, 51, 25, 35, 52syl22anc 838 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)𝑦) ∈ (𝑇 βŠ• π‘ˆ))
5419, 44, 45, 6lssvscl 20559 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝐿) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ ((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦) ∈ 𝑇)) β†’ (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ∈ 𝑇)
5524, 26, 25, 36, 54syl22anc 838 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ∈ 𝑇)
5619, 44, 45, 6lssvscl 20559 . . . . . . . 8 (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿) ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ ((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦) ∈ π‘ˆ)) β†’ (π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦)) ∈ π‘ˆ)
5724, 38, 25, 42, 56syl22anc 838 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦)) ∈ π‘ˆ)
581, 2, 3, 4, 30, 31, 32, 33, 17, 53, 55, 57pj1eq 19563 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘₯( ·𝑠 β€˜π‘Š)𝑦) = ((π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))(+gβ€˜π‘Š)(π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))) ↔ (((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ∧ ((π‘ˆπ‘ƒπ‘‡)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦)))))
5948, 58mpbid 231 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ (((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ∧ ((π‘ˆπ‘ƒπ‘‡)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘ˆπ‘ƒπ‘‡)β€˜π‘¦))))
6059simpld 496 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑇 βŠ• π‘ˆ))) β†’ ((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)))
6160ralrimivva 3201 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (𝑇 βŠ• π‘ˆ)((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)))
628, 50sseldd 3983 . . . . . 6 (πœ‘ β†’ (𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜π‘Š))
63 eqid 2733 . . . . . . 7 (π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) = (π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))
6463subgbas 19005 . . . . . 6 ((𝑇 βŠ• π‘ˆ) ∈ (SubGrpβ€˜π‘Š) β†’ (𝑇 βŠ• π‘ˆ) = (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))))
6562, 64syl 17 . . . . 5 (πœ‘ β†’ (𝑇 βŠ• π‘ˆ) = (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))))
6665raleqdv 3326 . . . 4 (πœ‘ β†’ (βˆ€π‘¦ ∈ (𝑇 βŠ• π‘ˆ)((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))))
6766ralbidv 3178 . . 3 (πœ‘ β†’ (βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (𝑇 βŠ• π‘ˆ)((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦))))
6861, 67mpbid 231 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)))
6963, 6lsslmod 20564 . . . 4 ((π‘Š ∈ LMod ∧ (𝑇 βŠ• π‘ˆ) ∈ 𝐿) β†’ (π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) ∈ LMod)
705, 50, 69syl2anc 585 . . 3 (πœ‘ β†’ (π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) ∈ LMod)
71 ovex 7439 . . . . 5 (𝑇 βŠ• π‘ˆ) ∈ V
7263, 19resssca 17285 . . . . 5 ((𝑇 βŠ• π‘ˆ) ∈ V β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))))
7371, 72ax-mp 5 . . . 4 (Scalarβ€˜π‘Š) = (Scalarβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))
74 eqid 2733 . . . 4 (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))) = (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))
7563, 44ressvsca 17286 . . . . 5 ((𝑇 βŠ• π‘ˆ) ∈ V β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ))))
7671, 75ax-mp 5 . . . 4 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))
7773, 19, 45, 74, 76, 44islmhm3 20632 . . 3 (((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) ∈ LMod ∧ π‘Š ∈ LMod) β†’ ((π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) LMHom π‘Š) ↔ ((π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) GrpHom π‘Š) ∧ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)))))
7870, 5, 77syl2anc 585 . 2 (πœ‘ β†’ ((π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) LMHom π‘Š) ↔ ((π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) GrpHom π‘Š) ∧ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ (Baseβ€˜(π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)))((π‘‡π‘ƒπ‘ˆ)β€˜(π‘₯( ·𝑠 β€˜π‘Š)𝑦)) = (π‘₯( ·𝑠 β€˜π‘Š)((π‘‡π‘ƒπ‘ˆ)β€˜π‘¦)))))
7918, 20, 68, 78mpbir3and 1343 1 (πœ‘ β†’ (π‘‡π‘ƒπ‘ˆ) ∈ ((π‘Š β†Ύs (𝑇 βŠ• π‘ˆ)) LMHom π‘Š))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   ∩ cin 3947   βŠ† wss 3948  {csn 4628  β€˜cfv 6541  (class class class)co 7406  Basecbs 17141   β†Ύs cress 17170  +gcplusg 17194  Scalarcsca 17197   ·𝑠 cvsca 17198  0gc0g 17382  SubGrpcsubg 18995   GrpHom cghm 19084  Cntzccntz 19174  LSSumclsm 19497  proj1cpj1 19498  Abelcabl 19644  LModclmod 20464  LSubSpclss 20535   LMHom clmhm 20623
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  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-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-ress 17171  df-plusg 17207  df-sca 17210  df-vsca 17211  df-0g 17384  df-mgm 18558  df-sgrp 18607  df-mnd 18623  df-submnd 18669  df-grp 18819  df-minusg 18820  df-sbg 18821  df-subg 18998  df-ghm 19085  df-cntz 19176  df-lsm 19499  df-pj1 19500  df-cmn 19645  df-abl 19646  df-mgp 19983  df-ur 20000  df-ring 20052  df-lmod 20466  df-lss 20536  df-lmhm 20626
This theorem is referenced by:  pj1lmhm2  20705  pjff  21259
  Copyright terms: Public domain W3C validator