lsppratlem3 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lsppratlem3		Structured version Visualization version GIF version

Theorem lsppratlem3 21081

Description: Lemma for lspprat 21085. In the first case of lsppratlem1 21079, since 𝑥 ∉ (𝑁‘∅), also 𝑌 ∈ (𝑁‘{𝑥}), and since 𝑦 ∈ (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘{𝑋, 𝑥}) and 𝑦 ∉ (𝑁‘{𝑥}), we have 𝑋 ∈ (𝑁‘{𝑥, 𝑦}) as desired. (Contributed by NM, 29-Aug-2014.)

**Hypotheses**
Ref	Expression
lspprat.v	⊢ 𝑉 = (Base‘𝑊)
lspprat.s	⊢ 𝑆 = (LSubSp‘𝑊)
lspprat.n	⊢ 𝑁 = (LSpan‘𝑊)
lspprat.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspprat.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lspprat.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lspprat.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
lspprat.p	⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))
lsppratlem1.o	⊢ 0 = (0_g‘𝑊)
lsppratlem1.x2	⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))
lsppratlem1.y2	⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
lsppratlem3.x3	⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
lsppratlem3	⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

**Proof of Theorem lsppratlem3**
Step	Hyp	Ref	Expression
1		lspprat.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ LVec)
2		lveclmod 21035	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspprat.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5	4	snssd 4756	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
6		lspprat.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7		lspprat.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
8	6, 7	lspssv 20911	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
9	3, 5, 8	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
10		lsppratlem3.x3	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))
11	9, 10	sseldd 3930	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
12	11	snssd 4756	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ 𝑉)
13		lspprat.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14		lspprat.p	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))
15	14	pssssd 4045	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌}))
16	13	snssd 4756	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
17	12, 16	unssd 4137	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ 𝑉)
18		lspprat.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊)
19	6, 18, 7	lspcl 20904	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
20	3, 17, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
21		df-pr 4574	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
22	6, 7	lspssid 20913	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
23	3, 17, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
24	23	unssbd 4139	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
25		ssun1 4123	. . . . . . . . . . . . . 14 ⊢ {𝑥} ⊆ ({𝑥} ∪ {𝑋})
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑥} ⊆ ({𝑥} ∪ {𝑋}))
27	6, 7	lspss 20912	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉 ∧ {𝑥} ⊆ ({𝑥} ∪ {𝑋})) → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
28	3, 17, 26, 27	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
29		0ss 4345	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ 𝑉
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ⊆ 𝑉)
31		uncom 4103	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ {𝑌}) = ({𝑌} ∪ ∅)
32		un0 4339	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑌} ∪ ∅) = {𝑌}
33	31, 32	eqtri 2754	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ {𝑌}) = {𝑌}
34	33	fveq2i 6820	. . . . . . . . . . . . . . . 16 ⊢ (𝑁‘(∅ ∪ {𝑌})) = (𝑁‘{𝑌})
35	10, 34	eleqtrrdi 2842	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘(∅ ∪ {𝑌})))
36		lsppratlem1.x2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))
37	36	eldifbd 3910	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑥 ∈ { 0 })
38		lsppratlem1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0_g‘𝑊)
39	38, 7	lsp0 20937	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
40	3, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘∅) = { 0 })
41	37, 40	neleqtrrd 2854	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘∅))
42	35, 41	eldifd 3908	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))
43	6, 18, 7	lspsolv 21075	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (∅ ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))) → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
44	1, 30, 4, 42, 43	syl13anc 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
45		uncom 4103	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {𝑥}) = ({𝑥} ∪ ∅)
46		un0 4339	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∪ ∅) = {𝑥}
47	45, 46	eqtri 2754	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ {𝑥}) = {𝑥}
48	47	fveq2i 6820	. . . . . . . . . . . . 13 ⊢ (𝑁‘(∅ ∪ {𝑥})) = (𝑁‘{𝑥})
49	44, 48	eleqtrdi 2841	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥}))
50	28, 49	sseldd 3930	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘({𝑥} ∪ {𝑋})))
51	50	snssd 4756	. . . . . . . . . 10 ⊢ (𝜑 → {𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
52	24, 51	unssd 4137	. . . . . . . . 9 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
53	21, 52	eqsstrid 3968	. . . . . . . 8 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
54	18, 7	lspssp 20916	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋}))) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
55	3, 20, 53, 54	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
56	15, 55	sstrd 3940	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
57	56	ssdifd 4090	. . . . 5 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
58		lsppratlem1.y2	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
59	57, 58	sseldd 3930	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
60	6, 18, 7	lspsolv 21075	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ ({𝑥} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
61	1, 12, 13, 59, 60	syl13anc 1374	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
62		df-pr 4574	. . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
63	62	fveq2i 6820	. . 3 ⊢ (𝑁‘{𝑥, 𝑦}) = (𝑁‘({𝑥} ∪ {𝑦}))
64	61, 63	eleqtrrdi 2842	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))
65		lspprat.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
66	6, 18	lssss 20864	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
67	65, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
68	67	ssdifssd 4092	. . . . . 6 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ 𝑉)
69	68, 58	sseldd 3930	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
70	11, 69	prssd 4769	. . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉)
71		snsspr1 4761	. . . . 5 ⊢ {𝑥} ⊆ {𝑥, 𝑦}
72	71	a1i 11	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ {𝑥, 𝑦})
73	6, 7	lspss 20912	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉 ∧ {𝑥} ⊆ {𝑥, 𝑦}) → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
74	3, 70, 72, 73	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
75	74, 49	sseldd 3930	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))
76	64, 75	jca 511	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 ⊊ wpss 3898 ∅c0 4278 {csn 4571 {cpr 4573 ‘cfv 6476 Basecbs 17115 0_gc0g 17338 LModclmod 20788 LSubSpclss 20859 LSpanclspn 20899 LVecclvec 21031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-nn 12121 df-2 12183 df-3 12184 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-0g 17340 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-grp 18844 df-minusg 18845 df-sbg 18846 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-drng 20641 df-lmod 20790 df-lss 20860 df-lsp 20900 df-lvec 21032

This theorem is referenced by: lsppratlem5 21083

W3C validator