Theorem lspextmo 21332

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 2782	. . . 4 ⊢ (((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
2		inss1 4133	. . . . . . . . 9 ⊢ (𝑔 ∩ ℎ) ⊆ 𝑔
3		dmss 5883	. . . . . . . . 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 2760	. . . . . . . . . . . . 13 ⊢ (Base‘𝑇) = (Base‘𝑇)
7	5, 6	lmhmf 21310	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑔:𝐵⟶(Base‘𝑇))
8	7	ad2antrl 741	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔:𝐵⟶(Base‘𝑇))
9	8	ffnd 6710	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑔 Fn 𝐵)
10	9	adantrr 730	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑔 Fn 𝐵)
11	10	fndmd 6644	. . . . . . . 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 3950	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → dom (𝑔 ∩ ℎ) ⊆ 𝐵)
13		simplr 781	. . . . . . . 8 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → (𝐾‘𝑋) = 𝐵)
14		lmhmlmod1 21309	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
15	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → 𝑆 ∈ LMod)
16	15	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ))) → 𝑆 ∈ LMod)
17		eqid 2760	. . . . . . . . . . 11 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
18	17	lmhmeql 21331	. . . . . . . . . 10 ⊢ ((𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇)) → dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆))
19	18	ad2antrl 741	. . . . . . . . 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 785	. . . . . . . . 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 21264	. . . . . . . . 9 ⊢ ((𝑆 ∈ LMod ∧ dom (𝑔 ∩ ℎ) ∈ (LSubSp‘𝑆) ∧ 𝑋 ⊆ dom (𝑔 ∩ ℎ)) → (𝐾‘𝑋) ⊆ dom (𝑔 ∩ ℎ))
23	16, 19, 20, 22	syl3anc 1398	. . . . . . . 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 3943	. . . . . . 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 3925	. . . . . 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 462	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → (𝑋 ⊆ dom (𝑔 ∩ ℎ) → dom (𝑔 ∩ ℎ) = 𝐵))
27		simprr 785	. . . . . . 7 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ ∈ (𝑆 LMHom 𝑇))
28	5, 6	lmhmf 21310	. . . . . . 7 ⊢ (ℎ ∈ (𝑆 LMHom 𝑇) → ℎ:𝐵⟶(Base‘𝑇))
29		ffn 6709	. . . . . . 7 ⊢ (ℎ:𝐵⟶(Base‘𝑇) → ℎ Fn 𝐵)
30	27, 28, 29	3syl 19	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ℎ Fn 𝐵)
31		simpll 779	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → 𝑋 ⊆ 𝐵)
32		fnreseql 7047	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵 ∧ 𝑋 ⊆ 𝐵) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
33	9, 30, 31, 32	syl3anc 1398	. . . . 5 ⊢ (((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) ∧ (𝑔 ∈ (𝑆 LMHom 𝑇) ∧ ℎ ∈ (𝑆 LMHom 𝑇))) → ((𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝑔 ∩ ℎ)))
34		fneqeql 7045	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
35	9, 30, 34	syl2anc 596	. . . . 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 35	. . 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 3205	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
39		reseq1 5963	. . . 4 ⊢ (𝑔 = ℎ → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
40	39	eqeq1d 2762	. . 3 ⊢ (𝑔 = ℎ → ((𝑔 ↾ 𝑋) = 𝐹 ↔ (ℎ ↾ 𝑋) = 𝐹))
41	40	rmo4 3687	. 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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃*wrmo 3364   ⊆ wss 3895   ∩ cin 4089  dom cdm 5650   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  LModclmod 21136  LSubSpclss 21207  LSpanclspn 21247   LMHom clmhm 21295

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-mhm 18979  df-submnd 18980  df-grp 19148  df-minusg 19149  df-sbg 19150  df-subg 19334  df-ghm 19429  df-mgp 20362  df-ur 20409  df-ring 20462  df-lmod 21138  df-lss 21208  df-lsp 21248  df-lmhm 21298

This theorem is used by:  frlmup4  22108