Theorem lspextmo 18877

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	⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹)

Distinct variable groups:   𝐵,𝑔   𝑔,𝐹   𝑔,𝐾   𝑆,𝑔   𝑇,𝑔   𝑔,𝑋

Proof of Theorem lspextmo

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqtr3 2631	. . . 4 ⊢ (((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
2		inss1 3795	. . . . . . . . 9 ⊢ (𝑔 ∩ ℎ) ⊆ 𝑔
3		dmss 5245	. . . . . . . . 9 ⊢ ((𝑔 ∩ ℎ) ⊆ 𝑔 → dom (𝑔 ∩ ℎ) ⊆ dom 𝑔)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ dom (𝑔 ∩ ℎ) ⊆ dom 𝑔
5		lspextmo.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝑆)
6		eqid 2610	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	lmhmf 18855	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
8	7	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
9		ffn 5958	. . . . . . . . . . 11 ⊢ (𝑔:𝐵⟶(Base‘𝑇) → 𝑔 Fn 𝐵)
10	8, 9	syl 17	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔 Fn 𝐵)
11	10	adantrr 749	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑔 Fn 𝐵)
12		fndm 5904	. . . . . . . . 9 ⊢ (𝑔 Fn 𝐵 → dom 𝑔 = 𝐵)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom 𝑔 = 𝐵)
14	4, 13	syl5sseq 3616	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) ⊆ 𝐵)
15		simplr 788	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → (𝐾‘𝑋) = 𝐵)
16		lmhmlmod1 18854	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
18	17	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑆 ∈ LMod)
19		eqid 2610	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
20	19	lmhmeql 18876	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆))
21	20	ad2antrl 760	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆))
22		simprr 792	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑋 ⊆ dom (𝑔 ∩ ℎ))
23		lspextmo.k	. . . . . . . . . 10 ⊢ 𝐾 = (LSpan‘𝑆)
24	19, 23	lspssp 18809	. . . . . . . . 9 ⊢ ((𝑆 ∈ LMod ∧ dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ)) → (𝐾‘𝑋) ⊆ dom (𝑔 ∩ ℎ))
25	18, 21, 22, 24	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → (𝐾‘𝑋) ⊆ dom (𝑔 ∩ ℎ))
26	15, 25	eqsstr3d 3603	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝐵 ⊆ dom (𝑔 ∩ ℎ))
27	14, 26	eqssd 3585	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) = 𝐵)
28	27	expr 641	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (𝑋 ⊆ dom (𝑔 ∩ ℎ) → dom (𝑔 ∩ ℎ) = 𝐵))
29		simprr 792	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ ∈ (𝑆 LMHom 𝑇))
30	5, 6	lmhmf 18855	. . . . . . 7 ⊢ (ℎ ∈ (𝑆 LMHom 𝑇) → ℎ:𝐵⟶(Base‘𝑇))
31		ffn 5958	. . . . . . 7 ⊢ (ℎ:𝐵⟶(Base‘𝑇) → ℎ Fn 𝐵)
32	29, 30, 31	3syl 18	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ Fn 𝐵)
33		simpll 786	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑋 ⊆ 𝐵)
34		fnreseql 6235	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵 ∧ 𝑋 ⊆ 𝐵) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
35	10, 32, 33, 34	syl3anc 1318	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
36		fneqeql 6233	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
37	10, 32, 36	syl2anc 691	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
38	28, 35, 37	3imtr4d 282	. . . 4 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) → 𝑔 = ℎ))
39	1, 38	syl5 33	. . 3 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
40	39	ralrimivva 2954	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
41		reseq1 5311	. . . 4 ⊢ (𝑔 = ℎ → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
42	41	eqeq1d 2612	. . 3 ⊢ (𝑔 = ℎ → ((𝑔 ↾ 𝑋) = 𝐹 ↔ (ℎ ↾ 𝑋) = 𝐹))
43	42	rmo4 3366	. 2 ⊢ (∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹 ↔ ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
44	40, 43	sylibr 223	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃*wrmo 2899   ∩ cin 3539   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  LModclmod 18686  LSubSpclss 18753  LSpanclspn 18792   LMHom clmhm 18840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lmhm 18843

This theorem is referenced by:  frlmup4  19959