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Theorem lsppratlem3 21250

Description: Lemma for lspprat 21254. In the first case of lsppratlem1 21248, 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 21204	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspprat.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5	4	snssd 4757	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
6		lspprat.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7		lspprat.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
8	6, 7	lspssv 21081	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
9	3, 5, 8	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
10		lsppratlem3.x3	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))
11	9, 10	sseldd 3946	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
12	11	snssd 4757	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ 𝑉)
13		lspprat.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14		lspprat.p	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))
15	14	pssssd 4062	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌}))
16	13	snssd 4757	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
17	12, 16	unssd 4153	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ 𝑉)
18		lspprat.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊)
19	6, 18, 7	lspcl 21074	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
20	3, 17, 19	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
21		df-pr 4597	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
22	6, 7	lspssid 21083	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
23	3, 17, 22	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
24	23	unssbd 4155	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
25		ssun1 4139	. . . . . . . . . . . . . 14 ⊢ {𝑥} ⊆ ({𝑥} ∪ {𝑋})
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑥} ⊆ ({𝑥} ∪ {𝑋}))
27	6, 7	lspss 21082	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉 ∧ {𝑥} ⊆ ({𝑥} ∪ {𝑋})) → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
28	3, 17, 26, 27	syl3anc 1396	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
29		0ss 4364	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ 𝑉
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ⊆ 𝑉)
31		uncom 4120	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ {𝑌}) = ({𝑌} ∪ ∅)
32		un0 4358	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑌} ∪ ∅) = {𝑌}
33	31, 32	eqtri 2792	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ {𝑌}) = {𝑌}
34	33	fveq2i 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑁‘(∅ ∪ {𝑌})) = (𝑁‘{𝑌})
35	10, 34	eleqtrrdi 2880	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘(∅ ∪ {𝑌})))
36		lsppratlem1.x2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))
37	36	eldifbd 3926	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑥 ∈ { 0 })
38		lsppratlem1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0_g‘𝑊)
39	38, 7	lsp0 21107	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
40	3, 39	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘∅) = { 0 })
41	37, 40	neleqtrrd 2892	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘∅))
42	35, 41	eldifd 3924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))
43	6, 18, 7	lspsolv 21244	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (∅ ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))) → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
44	1, 30, 4, 42, 43	syl13anc 1397	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
45		uncom 4120	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {𝑥}) = ({𝑥} ∪ ∅)
46		un0 4358	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∪ ∅) = {𝑥}
47	45, 46	eqtri 2792	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ {𝑥}) = {𝑥}
48	47	fveq2i 6885	. . . . . . . . . . . . 13 ⊢ (𝑁‘(∅ ∪ {𝑥})) = (𝑁‘{𝑥})
49	44, 48	eleqtrdi 2879	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥}))
50	28, 49	sseldd 3946	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘({𝑥} ∪ {𝑋})))
51	50	snssd 4757	. . . . . . . . . 10 ⊢ (𝜑 → {𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
52	24, 51	unssd 4153	. . . . . . . . 9 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
53	21, 52	eqsstrid 3983	. . . . . . . 8 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
54	18, 7	lspssp 21086	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋}))) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
55	3, 20, 53, 54	syl3anc 1396	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
56	15, 55	sstrd 3955	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
57	56	ssdifd 4107	. . . . 5 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
58		lsppratlem1.y2	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
59	57, 58	sseldd 3946	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
60	6, 18, 7	lspsolv 21244	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ ({𝑥} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
61	1, 12, 13, 59, 60	syl13anc 1397	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
62		df-pr 4597	. . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
63	62	fveq2i 6885	. . 3 ⊢ (𝑁‘{𝑥, 𝑦}) = (𝑁‘({𝑥} ∪ {𝑦}))
64	61, 63	eleqtrrdi 2880	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))
65		lspprat.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
66	6, 18	lssss 21034	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
67	65, 66	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
68	67	ssdifssd 4109	. . . . . 6 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ 𝑉)
69	68, 58	sseldd 3946	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
70	11, 69	prssd 4792	. . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉)
71		snsspr1 4784	. . . . 5 ⊢ {𝑥} ⊆ {𝑥, 𝑦}
72	71	a1i 11	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ {𝑥, 𝑦})
73	6, 7	lspss 21082	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉 ∧ {𝑥} ⊆ {𝑥, 𝑦}) → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
74	3, 70, 72, 73	syl3anc 1396	. . 3 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
75	74, 49	sseldd 3946	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))
76	64, 75	jca 520	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 ⊊ wpss 3914 ∅c0 4294 {csn 4594 {cpr 4596 ‘cfv 6537 Basecbs 17268 0_gc0g 17491 LModclmod 20958 LSubSpclss 21029 LSpanclspn 21069 LVecclvec 21200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-0g 17493 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-grp 19002 df-minusg 19003 df-sbg 19004 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-drng 20814 df-lmod 20960 df-lss 21030 df-lsp 21070 df-lvec 21201

This theorem is referenced by: lsppratlem5 21252

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