MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspextmo Structured version   Visualization version   GIF version

Theorem lspextmo 19104
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 2672 . . . 4 (((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → (𝑔𝑋) = (𝑋))
2 inss1 3866 . . . . . . . . 9 (𝑔) ⊆ 𝑔
3 dmss 5355 . . . . . . . . 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 2651 . . . . . . . . . . . . 13 (Base‘𝑇) = (Base‘𝑇)
75, 6lmhmf 19082 . . . . . . . . . . . 12 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
87ad2antrl 764 . . . . . . . . . . 11 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
9 ffn 6083 . . . . . . . . . . 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 753 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑔 Fn 𝐵)
12 fndm 6028 . . . . . . . . 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 3686 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ⊆ 𝐵)
15 simplr 807 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) = 𝐵)
16 lmhmlmod1 19081 . . . . . . . . . . 11 (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1716adantr 480 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
1817ad2antrl 764 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝑆 ∈ LMod)
19 eqid 2651 . . . . . . . . . . 11 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2019lmhmeql 19103 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔) ∈ (LSubSp‘𝑆))
2120ad2antrl 764 . . . . . . . . 9 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → dom (𝑔) ∈ (LSubSp‘𝑆))
22 simprr 811 . . . . . . . . 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 19036 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ dom (𝑔) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔)) → (𝐾𝑋) ⊆ dom (𝑔))
2518, 21, 22, 24syl3anc 1366 . . . . . . . 8 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → (𝐾𝑋) ⊆ dom (𝑔))
2615, 25eqsstr3d 3673 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔))) → 𝐵 ⊆ dom (𝑔))
2714, 26eqssd 3653 . . . . . 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 811 . . . . . . 7 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ∈ (𝑆 LMHom 𝑇))
305, 6lmhmf 19082 . . . . . . 7 ( ∈ (𝑆 LMHom 𝑇) → :𝐵⟶(Base‘𝑇))
31 ffn 6083 . . . . . . 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 805 . . . . . 6 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → 𝑋𝐵)
34 fnreseql 6367 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵𝑋𝐵) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
3510, 32, 33, 34syl3anc 1366 . . . . 5 (((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ∈ (𝑆 LMHom 𝑇))) → ((𝑔𝑋) = (𝑋) ↔ 𝑋 ⊆ dom (𝑔)))
36 fneqeql 6365 . . . . . 6 ((𝑔 Fn 𝐵 Fn 𝐵) → (𝑔 = ↔ dom (𝑔) = 𝐵))
3710, 32, 36syl2anc 694 . . . . 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 3000 . 2 ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ ∈ (𝑆 LMHom 𝑇)(((𝑔𝑋) = 𝐹 ∧ (𝑋) = 𝐹) → 𝑔 = ))
41 reseq1 5422 . . . 4 (𝑔 = → (𝑔𝑋) = (𝑋))
4241eqeq1d 2653 . . 3 (𝑔 = → ((𝑔𝑋) = 𝐹 ↔ (𝑋) = 𝐹))
4342rmo4 3432 . 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 383   = wceq 1523  wcel 2030  wral 2941  ∃*wrmo 2944  cin 3606  wss 3607  dom cdm 5143  cres 5145   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  LModclmod 18911  LSubSpclss 18980  LSpanclspn 19019   LMHom clmhm 19067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-ghm 17705  df-mgp 18536  df-ur 18548  df-ring 18595  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lmhm 19070
This theorem is referenced by:  frlmup4  20188
  Copyright terms: Public domain W3C validator