Theorem lspextmo 21055

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 4237	. . . . . . . . 9 ⊢ (𝑔 ∩ ℎ) ⊆ 𝑔
3		dmss 5913	. . . . . . . . 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 21033	. . . . . . . . . . . 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 6737	. . . . . . . . . 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 6673	. . . . . . . 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 4026	. . . . . . 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 21032	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
15	14	adantr 480	. . . . . . . . . 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 21054	. . . . . . . . . 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 20986	. . . . . . . . 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 4019	. . . . . . 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 4001	. . . . . 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 456	. . . . 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 21033	. . . . . . 7 ⊢ (ℎ ∈ (𝑆 LMHom 𝑇) → ℎ:𝐵⟶(Base‘𝑇))
29		ffn 6736	. . . . . . 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 7068	. . . . . 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 7066	. . . . . 6 ⊢ ((𝑔 Fn 𝐵 ∧ ℎ Fn 𝐵) → (𝑔 = ℎ ↔ dom (𝑔 ∩ ℎ) = 𝐵))
35	9, 30, 34	syl2anc 584	. . . . 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 294	. . . 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 3202	. 2 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∀𝑔 ∈ (𝑆 LMHom 𝑇)∀ℎ ∈ (𝑆 LMHom 𝑇)(((𝑔 ↾ 𝑋) = 𝐹 ∧ (ℎ ↾ 𝑋) = 𝐹) → 𝑔 = ℎ))
39		reseq1 5991	. . . 4 ⊢ (𝑔 = ℎ → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
40	39	eqeq1d 2739	. . 3 ⊢ (𝑔 = ℎ → ((𝑔 ↾ 𝑋) = 𝐹 ↔ (ℎ ↾ 𝑋) = 𝐹))
41	40	rmo4 3736	. 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 234	1 ⊢ ((𝑋 ⊆ 𝐵 ∧ (𝐾‘𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔 ↾ 𝑋) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃*wrmo 3379   ∩ cin 3950   ⊆ wss 3951  dom cdm 5685   ↾ cres 5687   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  LModclmod 20858  LSubSpclss 20929  LSpanclspn 20969   LMHom clmhm 21018

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-mgp 20138  df-ur 20179  df-ring 20232  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021

This theorem is referenced by:  frlmup4  21821