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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
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