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Theorem lspextmo 20666
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 2758 . . . 4 (((𝑔 β†Ύ 𝑋) = 𝐹 ∧ (β„Ž β†Ύ 𝑋) = 𝐹) β†’ (𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋))
2 inss1 4228 . . . . . . . . 9 (𝑔 ∩ β„Ž) βŠ† 𝑔
3 dmss 5902 . . . . . . . . 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 2732 . . . . . . . . . . . . 13 (Baseβ€˜π‘‡) = (Baseβ€˜π‘‡)
75, 6lmhmf 20644 . . . . . . . . . . . 12 (𝑔 ∈ (𝑆 LMHom 𝑇) β†’ 𝑔:𝐡⟢(Baseβ€˜π‘‡))
87ad2antrl 726 . . . . . . . . . . 11 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑔:𝐡⟢(Baseβ€˜π‘‡))
98ffnd 6718 . . . . . . . . . 10 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑔 Fn 𝐡)
109adantrr 715 . . . . . . . . 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 6654 . . . . . . . 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 4034 . . . . . . 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 767 . . . . . . . 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 20643 . . . . . . . . . . 11 (𝑔 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
1514adantr 481 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) β†’ 𝑆 ∈ LMod)
1615ad2antrl 726 . . . . . . . . 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 2732 . . . . . . . . . . 11 (LSubSpβ€˜π‘†) = (LSubSpβ€˜π‘†)
1817lmhmeql 20665 . . . . . . . . . 10 ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇)) β†’ dom (𝑔 ∩ β„Ž) ∈ (LSubSpβ€˜π‘†))
1918ad2antrl 726 . . . . . . . . 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 771 . . . . . . . . 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 20598 . . . . . . . . 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 1371 . . . . . . . 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 4021 . . . . . . 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 3999 . . . . . 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 457 . . . . 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 771 . . . . . . 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 20644 . . . . . . 7 (β„Ž ∈ (𝑆 LMHom 𝑇) β†’ β„Ž:𝐡⟢(Baseβ€˜π‘‡))
29 ffn 6717 . . . . . . 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 765 . . . . . 6 (((𝑋 βŠ† 𝐡 ∧ (πΎβ€˜π‘‹) = 𝐡) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ β„Ž ∈ (𝑆 LMHom 𝑇))) β†’ 𝑋 βŠ† 𝐡)
32 fnreseql 7049 . . . . . 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 1371 . . . . 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 7047 . . . . . 6 ((𝑔 Fn 𝐡 ∧ β„Ž Fn 𝐡) β†’ (𝑔 = β„Ž ↔ dom (𝑔 ∩ β„Ž) = 𝐡))
359, 30, 34syl2anc 584 . . . . 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 3200 . 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 5975 . . . 4 (𝑔 = β„Ž β†’ (𝑔 β†Ύ 𝑋) = (β„Ž β†Ύ 𝑋))
4039eqeq1d 2734 . . 3 (𝑔 = β„Ž β†’ ((𝑔 β†Ύ 𝑋) = 𝐹 ↔ (β„Ž β†Ύ 𝑋) = 𝐹))
4140rmo4 3726 . 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒ*wrmo 3375   ∩ cin 3947   βŠ† wss 3948  dom cdm 5676   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  LModclmod 20470  LSubSpclss 20541  LSpanclspn 20581   LMHom clmhm 20629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-ghm 19089  df-mgp 19987  df-ur 20004  df-ring 20057  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lmhm 20632
This theorem is referenced by:  frlmup4  21355
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