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Theorem lspextmo 20811
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 𝐡 = (Baseβ€˜π‘†)
lspextmo.k 𝐾 = (LSpanβ€˜π‘†)
Assertion
Ref Expression
lspextmo ((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) β†’ βˆƒ*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 β†Ύ 𝑋) = 𝐹)
Distinct variable groups:   𝐡,𝑔   𝑔,𝐹   𝑔,𝐾   𝑆,𝑔   𝑇,𝑔   𝑔,𝑋
Proof of Theorem lspextmo
Dummy variable β„Ž is distinct from all other variables.
StepHypRef Expression
1 eqtr3 2756 . . . 4 (((𝑔 β†Ύ 𝑋) = 𝐹 ∧ (β„Ž β†Ύ 𝑋) = 𝐹) β†’ (𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋))
2 inss1 4227 . . . . . . . . 9 (𝑔 ∩ β„Ž) βŠ† 𝑔
3 dmss 5901 . . . . . . . . 9 ((𝑔 ∩ β„Ž) βŠ† 𝑔 β†’ dom (𝑔 ∩ β„Ž) βŠ† dom 𝑔)
42, 3ax-mp 5 . . . . . . . 8 dom (𝑔 ∩ β„Ž) βŠ† dom 𝑔
5 lspextmo.b . . . . . . . . . . . . 13 𝐡 = (Baseβ€˜π‘†)
6 eqid 2730 . . . . . . . . . . . . 13 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
75, 6lmhmf 20789 . . . . . . . . . . . 12 (𝑔 ∈ (𝑆 LMHom 𝑇) β†’ 𝑔:𝐡⟢(Baseβ€˜π‘‡))
87ad2antrl 724 . . . . . . . . . . 11 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑔:𝐡⟢(Baseβ€˜π‘‡))
98ffnd 6717 . . . . . . . . . 10 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑔 Fn 𝐡)
109adantrr 713 . . . . . . . . 9 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ 𝑔 Fn 𝐡)
1110fndmd 6653 . . . . . . . 8 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ dom 𝑔 = 𝐡)
124, 11sseqtrid 4033 . . . . . . 7 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ dom (𝑔 ∩ β„Ž) βŠ† 𝐡)
13 simplr 765 . . . . . . . 8 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ (πΎβ€˜π‘‹) = 𝐡)
14 lmhmlmod1 20788 . . . . . . . . . . 11 (𝑔 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
1514adantr 479 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) β†’ 𝑆 ∈ LMod)
1615ad2antrl 724 . . . . . . . . 9 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ 𝑆 ∈ LMod)
17 eqid 2730 . . . . . . . . . . 11 (LSubSpβ€˜π‘†) = (LSubSpβ€˜π‘†)
1817lmhmeql 20810 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝑔 ∩ β„Ž) ∈ (LSubSpβ€˜π‘†))
1918ad2antrl 724 . . . . . . . . 9 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ dom (𝑔 ∩ β„Ž) ∈ (LSubSpβ€˜π‘†))
20 simprr 769 . . . . . . . . 9 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))
21 lspextmo.k . . . . . . . . . 10 𝐾 = (LSpanβ€˜π‘†)
2217, 21lspssp 20743 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ dom (𝑔 ∩ β„Ž) ∈ (LSubSpβ€˜π‘†) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž)) β†’ (πΎβ€˜π‘‹) βŠ† dom (𝑔 ∩ β„Ž))
2316, 19, 20, 22syl3anc 1369 . . . . . . . 8 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ (πΎβ€˜π‘‹) βŠ† dom (𝑔 ∩ β„Ž))
2413, 23eqsstrrd 4020 . . . . . . 7 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ 𝐡 βŠ† dom (𝑔 ∩ β„Ž))
2512, 24eqssd 3998 . . . . . 6 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 βŠ† dom (𝑔 ∩ β„Ž))) β†’ dom (𝑔 ∩ β„Ž) = 𝐡)
2625expr 455 . . . . 5 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ (𝑋 βŠ† dom (𝑔 ∩ β„Ž) β†’ dom (𝑔 ∩ β„Ž) = 𝐡))
27 simprr 769 . . . . . . 7 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ β„Ž ∈ (𝑆 LMHom 𝑇))
285, 6lmhmf 20789 . . . . . . 7 (β„Ž ∈ (𝑆 LMHom 𝑇) β†’ β„Ž:𝐡⟢(Baseβ€˜π‘‡))
29 ffn 6716 . . . . . . 7 (β„Ž:𝐡⟢(Baseβ€˜π‘‡) β†’ β„Ž Fn 𝐡)
3027, 28, 293syl 18 . . . . . 6 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ β„Ž Fn 𝐡)
31 simpll 763 . . . . . 6 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑋 βŠ† 𝐡)
32 fnreseql 7048 . . . . . 6 ((𝑔 Fn 𝐡 ∧ β„Ž Fn 𝐡 ∧ 𝑋 βŠ† 𝐡) β†’ ((𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋) ↔ 𝑋 βŠ† dom (𝑔 ∩ β„Ž)))
339, 30, 31, 32syl3anc 1369 . . . . 5 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ ((𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋) ↔ 𝑋 βŠ† dom (𝑔 ∩ β„Ž)))
34 fneqeql 7046 . . . . . 6 ((𝑔 Fn 𝐡 ∧ β„Ž Fn 𝐡) β†’ (𝑔 = β„Ž ↔ dom (𝑔 ∩ β„Ž) = 𝐡))
359, 30, 34syl2anc 582 . . . . 5 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ (𝑔 = β„Ž ↔ dom (𝑔 ∩ β„Ž) = 𝐡))
3626, 33, 353imtr4d 293 . . . 4 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ ((𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋) β†’ 𝑔 = β„Ž))
371, 36syl5 34 . . 3 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ (((𝑔 β†Ύ 𝑋) = 𝐹 ∧ (β„Ž β†Ύ 𝑋) = 𝐹) β†’ 𝑔 = β„Ž))
3837ralrimivva 3198 . 2 ((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) β†’ βˆ€π‘” ∈ (𝑆 LMHom 𝑇)βˆ€β„Ž ∈ (𝑆 LMHom 𝑇)(((𝑔 β†Ύ 𝑋) = 𝐹 ∧ (β„Ž β†Ύ 𝑋) = 𝐹) β†’ 𝑔 = β„Ž))
39 reseq1 5974 . . . 4 (𝑔 = β„Ž β†’ (𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋))
4039eqeq1d 2732 . . 3 (𝑔 = β„Ž β†’ ((𝑔 β†Ύ 𝑋) = 𝐹 ↔ (β„Ž β†Ύ 𝑋) = 𝐹))
4140rmo4 3725 . 2 (βˆƒ*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 β†Ύ 𝑋) = 𝐹 ↔ βˆ€π‘” ∈ (𝑆 LMHom 𝑇)βˆ€β„Ž ∈ (𝑆 LMHom 𝑇)(((𝑔 β†Ύ 𝑋) = 𝐹 ∧ (β„Ž β†Ύ 𝑋) = 𝐹) β†’ 𝑔 = β„Ž))
4238, 41sylibr 233 1 ((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) β†’ βˆƒ*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 β†Ύ 𝑋) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒ*wrmo 3373   ∩ cin 3946   βŠ† wss 3947  dom cdm 5675   β†Ύ cres 5677   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  LModclmod 20614  LSubSpclss 20686  LSpanclspn 20726   LMHom clmhm 20774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-ghm 19128  df-mgp 20029  df-ur 20076  df-ring 20129  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lmhm 20777
This theorem is referenced by:  frlmup4  21575
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