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Theorem lsppratlem3 20669

Description: Lemma for lspprat 20673. In the first case of lsppratlem1 20667, 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 20624	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspprat.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5	4	snssd 4774	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
6		lspprat.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7		lspprat.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
8	6, 7	lspssv 20501	. . . . . . 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 3948	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
12	11	snssd 4774	. . . 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 4774	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
17	12, 16	unssd 4151	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ 𝑉)
18		lspprat.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊)
19	6, 18, 7	lspcl 20494	. . . . . . . . 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 4594	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
22	6, 7	lspssid 20503	. . . . . . . . . . . 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 4153	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
25		ssun1 4137	. . . . . . . . . . . . . 14 ⊢ {𝑥} ⊆ ({𝑥} ∪ {𝑋})
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑥} ⊆ ({𝑥} ∪ {𝑋}))
27	6, 7	lspss 20502	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉 ∧ {𝑥} ⊆ ({𝑥} ∪ {𝑋})) → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
28	3, 17, 26, 27	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
29		0ss 4361	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ 𝑉
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ⊆ 𝑉)
31		uncom 4118	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ {𝑌}) = ({𝑌} ∪ ∅)
32		un0 4355	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑌} ∪ ∅) = {𝑌}
33	31, 32	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ {𝑌}) = {𝑌}
34	33	fveq2i 6850	. . . . . . . . . . . . . . . 16 ⊢ (𝑁‘(∅ ∪ {𝑌})) = (𝑁‘{𝑌})
35	10, 34	eleqtrrdi 2843	. . . . . . . . . . . . . . 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 20527	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
40	3, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘∅) = { 0 })
41	37, 40	neleqtrrd 2855	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘∅))
42	35, 41	eldifd 3924	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))
43	6, 18, 7	lspsolv 20663	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (∅ ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))) → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
44	1, 30, 4, 42, 43	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
45		uncom 4118	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {𝑥}) = ({𝑥} ∪ ∅)
46		un0 4355	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∪ ∅) = {𝑥}
47	45, 46	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ {𝑥}) = {𝑥}
48	47	fveq2i 6850	. . . . . . . . . . . . 13 ⊢ (𝑁‘(∅ ∪ {𝑥})) = (𝑁‘{𝑥})
49	44, 48	eleqtrdi 2842	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥}))
50	28, 49	sseldd 3948	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘({𝑥} ∪ {𝑋})))
51	50	snssd 4774	. . . . . . . . . 10 ⊢ (𝜑 → {𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
52	24, 51	unssd 4151	. . . . . . . . 9 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
53	21, 52	eqsstrid 3995	. . . . . . . 8 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
54	18, 7	lspssp 20506	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋}))) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
55	3, 20, 53, 54	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
56	15, 55	sstrd 3957	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
57	56	ssdifd 4105	. . . . 5 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
58		lsppratlem1.y2	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
59	57, 58	sseldd 3948	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
60	6, 18, 7	lspsolv 20663	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ ({𝑥} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
61	1, 12, 13, 59, 60	syl13anc 1372	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
62		df-pr 4594	. . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
63	62	fveq2i 6850	. . 3 ⊢ (𝑁‘{𝑥, 𝑦}) = (𝑁‘({𝑥} ∪ {𝑦}))
64	61, 63	eleqtrrdi 2843	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))
65		lspprat.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
66	6, 18	lssss 20454	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
67	65, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
68	67	ssdifssd 4107	. . . . . 6 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ 𝑉)
69	68, 58	sseldd 3948	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
70	11, 69	prssd 4787	. . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉)
71		snsspr1 4779	. . . . 5 ⊢ {𝑥} ⊆ {𝑥, 𝑦}
72	71	a1i 11	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ {𝑥, 𝑦})
73	6, 7	lspss 20502	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉 ∧ {𝑥} ⊆ {𝑥, 𝑦}) → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
74	3, 70, 72, 73	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
75	74, 49	sseldd 3948	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))
76	64, 75	jca 512	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 ⊊ wpss 3914 ∅c0 4287 {csn 4591 {cpr 4593 ‘cfv 6501 Basecbs 17094 0_gc0g 17335 LModclmod 20378 LSubSpclss 20449 LSpanclspn 20489 LVecclvec 20620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11116 ax-resscn 11117 ax-1cn 11118 ax-icn 11119 ax-addcl 11120 ax-addrcl 11121 ax-mulcl 11122 ax-mulrcl 11123 ax-mulcom 11124 ax-addass 11125 ax-mulass 11126 ax-distr 11127 ax-i2m1 11128 ax-1ne0 11129 ax-1rid 11130 ax-rnegex 11131 ax-rrecex 11132 ax-cnre 11133 ax-pre-lttri 11134 ax-pre-lttrn 11135 ax-pre-ltadd 11136 ax-pre-mulgt0 11137

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3448 df-sbc 3743 df-csb 3859 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11200 df-mnf 11201 df-xr 11202 df-ltxr 11203 df-le 11204 df-sub 11396 df-neg 11397 df-nn 12163 df-2 12225 df-3 12226 df-sets 17047 df-slot 17065 df-ndx 17077 df-base 17095 df-ress 17124 df-plusg 17160 df-mulr 17161 df-0g 17337 df-mgm 18511 df-sgrp 18560 df-mnd 18571 df-grp 18765 df-minusg 18766 df-sbg 18767 df-cmn 19578 df-abl 19579 df-mgp 19911 df-ur 19928 df-ring 19980 df-oppr 20063 df-dvdsr 20084 df-unit 20085 df-invr 20115 df-drng 20227 df-lmod 20380 df-lss 20450 df-lsp 20490 df-lvec 20621

This theorem is referenced by: lsppratlem5 20671

W3C validator