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Theorem lsppratlem3 21142

Description: Lemma for lspprat 21146. In the first case of lsppratlem1 21140, 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 21096	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspprat.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5	4	snssd 4718	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
6		lspprat.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7		lspprat.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
8	6, 7	lspssv 20973	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
9	3, 5, 8	syl2anc 590	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
10		lsppratlem3.x3	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))
11	9, 10	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
12	11	snssd 4718	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ 𝑉)
13		lspprat.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14		lspprat.p	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))
15	14	pssssd 4031	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌}))
16	13	snssd 4718	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
17	12, 16	unssd 4121	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ 𝑉)
18		lspprat.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊)
19	6, 18, 7	lspcl 20966	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
20	3, 17, 19	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
21		df-pr 4558	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
22	6, 7	lspssid 20975	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
23	3, 17, 22	syl2anc 590	. . . . . . . . . . 11 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
24	23	unssbd 4123	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
25		ssun1 4107	. . . . . . . . . . . . . 14 ⊢ {𝑥} ⊆ ({𝑥} ∪ {𝑋})
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑥} ⊆ ({𝑥} ∪ {𝑋}))
27	6, 7	lspss 20974	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉 ∧ {𝑥} ⊆ ({𝑥} ∪ {𝑋})) → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
28	3, 17, 26, 27	syl3anc 1379	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
29		0ss 4328	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ 𝑉
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ⊆ 𝑉)
31		uncom 4088	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ {𝑌}) = ({𝑌} ∪ ∅)
32		un0 4322	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑌} ∪ ∅) = {𝑌}
33	31, 32	eqtri 2762	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ {𝑌}) = {𝑌}
34	33	fveq2i 6830	. . . . . . . . . . . . . . . 16 ⊢ (𝑁‘(∅ ∪ {𝑌})) = (𝑁‘{𝑌})
35	10, 34	eleqtrrdi 2850	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘(∅ ∪ {𝑌})))
36		lsppratlem1.x2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))
37	36	eldifbd 3896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑥 ∈ { 0 })
38		lsppratlem1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0_g‘𝑊)
39	38, 7	lsp0 20999	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
40	3, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘∅) = { 0 })
41	37, 40	neleqtrrd 2862	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘∅))
42	35, 41	eldifd 3894	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))
43	6, 18, 7	lspsolv 21136	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (∅ ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))) → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
44	1, 30, 4, 42, 43	syl13anc 1380	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
45		uncom 4088	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {𝑥}) = ({𝑥} ∪ ∅)
46		un0 4322	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∪ ∅) = {𝑥}
47	45, 46	eqtri 2762	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ {𝑥}) = {𝑥}
48	47	fveq2i 6830	. . . . . . . . . . . . 13 ⊢ (𝑁‘(∅ ∪ {𝑥})) = (𝑁‘{𝑥})
49	44, 48	eleqtrdi 2849	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥}))
50	28, 49	sseldd 3916	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘({𝑥} ∪ {𝑋})))
51	50	snssd 4718	. . . . . . . . . 10 ⊢ (𝜑 → {𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
52	24, 51	unssd 4121	. . . . . . . . 9 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
53	21, 52	eqsstrid 3953	. . . . . . . 8 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
54	18, 7	lspssp 20978	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋}))) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
55	3, 20, 53, 54	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
56	15, 55	sstrd 3925	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
57	56	ssdifd 4075	. . . . 5 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
58		lsppratlem1.y2	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
59	57, 58	sseldd 3916	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
60	6, 18, 7	lspsolv 21136	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ ({𝑥} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
61	1, 12, 13, 59, 60	syl13anc 1380	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
62		df-pr 4558	. . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
63	62	fveq2i 6830	. . 3 ⊢ (𝑁‘{𝑥, 𝑦}) = (𝑁‘({𝑥} ∪ {𝑦}))
64	61, 63	eleqtrrdi 2850	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))
65		lspprat.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
66	6, 18	lssss 20926	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
67	65, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
68	67	ssdifssd 4077	. . . . . 6 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ 𝑉)
69	68, 58	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
70	11, 69	prssd 4753	. . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉)
71		snsspr1 4745	. . . . 5 ⊢ {𝑥} ⊆ {𝑥, 𝑦}
72	71	a1i 11	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ {𝑥, 𝑦})
73	6, 7	lspss 20974	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉 ∧ {𝑥} ⊆ {𝑥, 𝑦}) → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
74	3, 70, 72, 73	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
75	74, 49	sseldd 3916	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))
76	64, 75	jca 516	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 ⊊ wpss 3884 ∅c0 4261 {csn 4555 {cpr 4557 ‘cfv 6485 Basecbs 17170 0_gc0g 17393 LModclmod 20850 LSubSpclss 20921 LSpanclspn 20961 LVecclvec 21092

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093

This theorem is referenced by: lsppratlem5 21144

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