Theorem lspextmo 20093

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 2763	. . . 4 ⊢ (((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
2		inss1 4143	. . . . . . . . 9 ⊢ (𝑔 ∩ ℎ) ⊆ 𝑔
3		dmss 5771	. . . . . . . . 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 2737	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	lmhmf 20071	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
8	7	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
9	8	ffnd 6546	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔 Fn 𝐵)
10	9	adantrr 717	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑔 Fn 𝐵)
11	10	fndmd 6483	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom 𝑔 = 𝐵)
12	4, 11	sseqtrid 3953	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) ⊆ 𝐵)
13		simplr 769	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → (𝐾‘𝑋) = 𝐵)
14		lmhmlmod1 20070	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
15	14	adantr 484	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
16	15	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑆 ∈ LMod)
17		eqid 2737	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
18	17	lmhmeql 20092	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆))
19	18	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆))
20		simprr 773	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑋 ⊆ dom (𝑔 ∩ ℎ))
21		lspextmo.k	. . . . . . . . . 10 ⊢ 𝐾 = (LSpan‘𝑆)
22	17, 21	lspssp 20025	. . . . . . . . 9 ⊢ ((𝑆 ∈ LMod ∧ dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ)) → (𝐾‘𝑋) ⊆ dom (𝑔 ∩ ℎ))
23	16, 19, 20, 22	syl3anc 1373	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → (𝐾‘𝑋) ⊆ dom (𝑔 ∩ ℎ))
24	13, 23	eqsstrrd 3940	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝐵 ⊆ dom (𝑔 ∩ ℎ))
25	12, 24	eqssd 3918	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) = 𝐵)
26	25	expr 460	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (𝑋 ⊆ dom (𝑔 ∩ ℎ) → dom (𝑔 ∩ ℎ) = 𝐵))
27		simprr 773	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ ∈ (𝑆 LMHom 𝑇))
28	5, 6	lmhmf 20071	. . . . . . 7 ⊢ (ℎ ∈ (𝑆 LMHom 𝑇) → ℎ:𝐵⟶(Base‘𝑇))
29		ffn 6545	. . . . . . 7 ⊢ (ℎ:𝐵⟶(Base‘𝑇) → ℎ Fn 𝐵)
30	27, 28, 29	3syl 18	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ Fn 𝐵)
31		simpll 767	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑋 ⊆ 𝐵)
32		fnreseql 6868	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵 ∧ 𝑋 ⊆ 𝐵) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
33	9, 30, 31, 32	syl3anc 1373	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
34		fneqeql 6866	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
35	9, 30, 34	syl2anc 587	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
36	26, 33, 35	3imtr4d 297	. . . 4 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) → 𝑔 = ℎ))
37	1, 36	syl5 34	. . 3 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
38	37	ralrimivva 3112	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
39		reseq1 5845	. . . 4 ⊢ (𝑔 = ℎ → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
40	39	eqeq1d 2739	. . 3 ⊢ (𝑔 = ℎ → ((𝑔 ↾ 𝑋) = 𝐹 ↔ (ℎ ↾ 𝑋) = 𝐹))
41	40	rmo4 3643	. 2 ⊢ (∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹 ↔ ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
42	38, 41	sylibr 237	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃*wrmo 3064   ∩ cin 3865   ⊆ wss 3866  dom cdm 5551   ↾ cres 5553   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  Basecbs 16760  LModclmod 19899  LSubSpclss 19968  LSpanclspn 20008   LMHom clmhm 20056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-ghm 18620  df-mgp 19505  df-ur 19517  df-ring 19564  df-lmod 19901  df-lss 19969  df-lsp 20009  df-lmhm 20059

This theorem is referenced by:  frlmup4  20763