Theorem lspextmo 21241

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 4182	. . . . . . . . 9 ⊢ (𝑔 ∩ ℎ) ⊆ 𝑔
3		dmss 5886	. . . . . . . . 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 21219	. . . . . . . . . . . 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 6704	. . . . . . . . . 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 6638	. . . . . . . 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 3973	. . . . . . 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 21218	. . . . . . . . . . 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 21240	. . . . . . . . . 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 21173	. . . . . . . . 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 3966	. . . . . . 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 3948	. . . . . 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 21219	. . . . . . 7 ⊢ (ℎ ∈ (𝑆 LMHom 𝑇) → ℎ:𝐵⟶(Base‘𝑇))
29		ffn 6703	. . . . . . 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 7041	. . . . . 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 7039	. . . . . 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 5966	. . . 4 ⊢ (𝑔 = ℎ → (𝑔 ↾ 𝑋) = (ℎ ↾ 𝑋))
40	39	eqeq1d 2762	. . 3 ⊢ (𝑔 = ℎ → ((𝑔 ↾ 𝑋) = 𝐹 ↔ (ℎ ↾ 𝑋) = 𝐹))
41	40	rmo4 3688	. 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 1570   ∈ wcel 2145  ∀wral 3076  ∃*wrmo 3364   ∩ cin 3898   ⊆ wss 3899  dom cdm 5655   ↾ cres 5657   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  LModclmod 21045  LSubSpclss 21116  LSpanclspn 21156   LMHom clmhm 21204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11181  ax-resscn 11182  ax-1cn 11183  ax-icn 11184  ax-addcl 11185  ax-addrcl 11186  ax-mulcl 11187  ax-mulrcl 11188  ax-mulcom 11189  ax-addass 11190  ax-mulass 11191  ax-distr 11192  ax-i2m1 11193  ax-1ne0 11194  ax-1rid 11195  ax-rnegex 11196  ax-rrecex 11197  ax-cnre 11198  ax-pre-lttri 11199  ax-pre-lttrn 11200  ax-pre-ltadd 11201  ax-pre-mulgt0 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-er 8697  df-map 8829  df-en 8954  df-dom 8955  df-sdom 8956  df-pnf 11270  df-mnf 11271  df-xr 11272  df-ltxr 11273  df-le 11274  df-sub 11468  df-neg 11469  df-nn 12259  df-2 12328  df-sets 17257  df-slot 17275  df-ndx 17287  df-base 17303  df-ress 17324  df-plusg 17356  df-0g 17527  df-mgm 18731  df-sgrp 18822  df-mnd 18838  df-mhm 18892  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-subg 19247  df-ghm 19342  df-mgp 20275  df-ur 20322  df-ring 20375  df-lmod 21047  df-lss 21117  df-lsp 21157  df-lmhm 21207

This theorem is used by:  frlmup4  22015