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Theorem lspextmo 21030
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 2752 . . . 4 (((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → (𝑔𝑋) = (𝑋))
2 inss1 4227 . . . . . . . . 9 (𝑔) ⊆ 𝑔
3 dmss 5901 . . . . . . . . 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 2726 . . . . . . . . . . . . 13 (Base‘𝑇) = (Base‘𝑇)
75, 6lmhmf 21008 . . . . . . . . . . . 12 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
87ad2antrl 726 . . . . . . . . . . 11 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
98ffnd 6721 . . . . . . . . . 10 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑔 Fn 𝐵)
109adantrr 715 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑔 Fn 𝐵)
1110fndmd 6657 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom 𝑔 = 𝐵)
124, 11sseqtrid 4031 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ⊆ 𝐵)
13 simplr 767 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) = 𝐵)
14 lmhmlmod1 21007 . . . . . . . . . . 11 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1514adantr 479 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
1615ad2antrl 726 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑆 ∈ LMod)
17 eqid 2726 . . . . . . . . . . 11 (LSubSp‘𝑆) = (LSubSp‘𝑆)
1817lmhmeql 21029 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔) ∈ (LSubSp‘𝑆))
1918ad2antrl 726 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ∈ (LSubSp‘𝑆))
20 simprr 771 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑋 ⊆ dom (𝑔))
21 lspextmo.k . . . . . . . . . 10 𝐾 = (LSpan‘𝑆)
2217, 21lspssp 20961 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ dom (𝑔) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔)) → (𝐾𝑋) ⊆ dom (𝑔))
2316, 19, 20, 22syl3anc 1368 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) ⊆ dom (𝑔))
2413, 23eqsstrrd 4018 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝐵 ⊆ dom (𝑔))
2512, 24eqssd 3996 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) = 𝐵)
2625expr 455 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (𝑋 ⊆ dom (𝑔) → dom (𝑔) = 𝐵))
27 simprr 771 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ∈ (𝑆 LMHom 𝑇))
285, 6lmhmf 21008 . . . . . . 7 ( ∈ (𝑆 LMHom 𝑇) → :𝐵⟶(Base‘𝑇))
29 ffn 6720 . . . . . . 7 (:𝐵⟶(Base‘𝑇) → Fn 𝐵)
3027, 28, 293syl 18 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → Fn 𝐵)
31 simpll 765 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑋𝐵)
32 fnreseql 7053 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵𝑋𝐵) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
339, 30, 31, 32syl3anc 1368 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
34 fneqeql 7051 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵) → (𝑔 = ↔ dom (𝑔) = 𝐵))
359, 30, 34syl2anc 582 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (𝑔 = ↔ dom (𝑔) = 𝐵))
3626, 33, 353imtr4d 293 . . . 4 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ((𝑔𝑋) = (𝑋) → 𝑔 = ))
371, 36syl5 34 . . 3 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → (((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
3837ralrimivva 3191 . 2 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ ∈ (𝑆 LMHom 𝑇)(((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
39 reseq1 5975 . . . 4 (𝑔 = → (𝑔𝑋) = (𝑋))
4039eqeq1d 2728 . . 3 (𝑔 = → ((𝑔𝑋) = 𝐹 ↔ (𝑋) = 𝐹))
4140rmo4 3723 . 2 (∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹 ↔ ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ ∈ (𝑆 LMHom 𝑇)(((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
4238, 41sylibr 233 1 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  ∃*wrmo 3363  cin 3945  wss 3946  dom cdm 5674  cres 5676   Fn wfn 6541  wf 6542  cfv 6546  (class class class)co 7416  Basecbs 17208  LModclmod 20832  LSubSpclss 20904  LSpanclspn 20944   LMHom clmhm 20993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-mhm 18768  df-submnd 18769  df-grp 18926  df-minusg 18927  df-sbg 18928  df-subg 19113  df-ghm 19203  df-mgp 20114  df-ur 20161  df-ring 20214  df-lmod 20834  df-lss 20905  df-lsp 20945  df-lmhm 20996
This theorem is referenced by:  frlmup4  21795
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