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Theorem lspextmo 18975
Description: A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
Hypotheses
Ref Expression
lspextmo.b 𝐵 = (Base‘𝑆)
lspextmo.k 𝐾 = (LSpan‘𝑆)
Assertion
Ref Expression
lspextmo ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹)
Distinct variable groups:   𝐵,𝑔   𝑔,𝐹   𝑔,𝐾   𝑆,𝑔   𝑇,𝑔   𝑔,𝑋

Proof of Theorem lspextmo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2642 . . . 4 (((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → (𝑔𝑋) = (𝑋))
2 inss1 3811 . . . . . . . . 9 (𝑔) ⊆ 𝑔
3 dmss 5283 . . . . . . . . 9 ((𝑔) ⊆ 𝑔 → dom (𝑔) ⊆ dom 𝑔)
42, 3ax-mp 5 . . . . . . . 8 dom (𝑔) ⊆ dom 𝑔
5 lspextmo.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝑆)
6 eqid 2621 . . . . . . . . . . . . 13 (Base‘𝑇) = (Base‘𝑇)
75, 6lmhmf 18953 . . . . . . . . . . . 12 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
87ad2antrl 763 . . . . . . . . . . 11 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
9 ffn 6002 . . . . . . . . . . 11 (𝑔:𝐵⟶(Base‘𝑇) → 𝑔 Fn 𝐵)
108, 9syl 17 . . . . . . . . . 10 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑔 Fn 𝐵)
1110adantrr 752 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑔 Fn 𝐵)
12 fndm 5948 . . . . . . . . 9 (𝑔 Fn 𝐵 → dom 𝑔 = 𝐵)
1311, 12syl 17 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom 𝑔 = 𝐵)
144, 13syl5sseq 3632 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ⊆ 𝐵)
15 simplr 791 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) = 𝐵)
16 lmhmlmod1 18952 . . . . . . . . . . 11 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1716adantr 481 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
1817ad2antrl 763 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑆 ∈ LMod)
19 eqid 2621 . . . . . . . . . . 11 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2019lmhmeql 18974 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔) ∈ (LSubSp‘𝑆))
2120ad2antrl 763 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ∈ (LSubSp‘𝑆))
22 simprr 795 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑋 ⊆ dom (𝑔))
23 lspextmo.k . . . . . . . . . 10 𝐾 = (LSpan‘𝑆)
2419, 23lspssp 18907 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ dom (𝑔) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔)) → (𝐾𝑋) ⊆ dom (𝑔))
2518, 21, 22, 24syl3anc 1323 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) ⊆ dom (𝑔))
2615, 25eqsstr3d 3619 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝐵 ⊆ dom (𝑔))
2714, 26eqssd 3600 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) = 𝐵)
2827expr 642 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (𝑋 ⊆ dom (𝑔) → dom (𝑔) = 𝐵))
29 simprr 795 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ∈ (𝑆 LMHom 𝑇))
305, 6lmhmf 18953 . . . . . . 7 ( ∈ (𝑆 LMHom 𝑇) → :𝐵⟶(Base‘𝑇))
31 ffn 6002 . . . . . . 7 (:𝐵⟶(Base‘𝑇) → Fn 𝐵)
3229, 30, 313syl 18 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → Fn 𝐵)
33 simpll 789 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑋𝐵)
34 fnreseql 6283 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵𝑋𝐵) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
3510, 32, 33, 34syl3anc 1323 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
36 fneqeql 6281 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵) → (𝑔 = ↔ dom (𝑔) = 𝐵))
3710, 32, 36syl2anc 692 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (𝑔 = ↔ dom (𝑔) = 𝐵))
3828, 35, 373imtr4d 283 . . . 4 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ((𝑔𝑋) = (𝑋) → 𝑔 = ))
391, 38syl5 34 . . 3 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
4039ralrimivva 2965 . 2 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ ∈ (𝑆 LMHom 𝑇)(((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
41 reseq1 5350 . . . 4 (𝑔 = → (𝑔𝑋) = (𝑋))
4241eqeq1d 2623 . . 3 (𝑔 = → ((𝑔𝑋) = 𝐹 ↔ (𝑋) = 𝐹))
4342rmo4 3381 . 2 (∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹 ↔ ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ ∈ (𝑆 LMHom 𝑇)(((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
4440, 43sylibr 224 1 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  ∃*wrmo 2910  cin 3554  wss 3555  dom cdm 5074  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  LModclmod 18784  LSubSpclss 18851  LSpanclspn 18890   LMHom clmhm 18938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-subg 17512  df-ghm 17579  df-mgp 18411  df-ur 18423  df-ring 18470  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lmhm 18941
This theorem is referenced by:  frlmup4  20059
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