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 19149

Description: Lemma for lspprat 19153. In the first case of lsppratlem1 19147, 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 19106	. . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspprat.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
5	4	snssd 4340	. . . . . . 7 ⊢ (𝜑 → {𝑌} ⊆ 𝑉)
6		lspprat.v	. . . . . . . 8 ⊢ 𝑉 = (Base‘𝑊)
7		lspprat.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑊)
8	6, 7	lspssv 18983	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ {𝑌} ⊆ 𝑉) → (𝑁‘{𝑌}) ⊆ 𝑉)
9	3, 5, 8	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ 𝑉)
10		lsppratlem3.x3	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘{𝑌}))
11	9, 10	sseldd 3604	. . . . 5 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
12	11	snssd 4340	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ 𝑉)
13		lspprat.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
14		lspprat.p	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))
15	14	pssssd 3704	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘{𝑋, 𝑌}))
16	13	snssd 4340	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
17	12, 16	unssd 3789	. . . . . . . . 9 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ 𝑉)
18		lspprat.s	. . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊)
19	6, 18, 7	lspcl 18976	. . . . . . . . 9 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
20	3, 17, 19	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆)
21		df-pr 4180	. . . . . . . . 9 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌})
22	6, 7	lspssid 18985	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉) → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
23	3, 17, 22	syl2anc 693	. . . . . . . . . . 11 ⊢ (𝜑 → ({𝑥} ∪ {𝑋}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
24	23	unssbd 3791	. . . . . . . . . 10 ⊢ (𝜑 → {𝑋} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
25		ssun1 3776	. . . . . . . . . . . . . 14 ⊢ {𝑥} ⊆ ({𝑥} ∪ {𝑋})
26	25	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → {𝑥} ⊆ ({𝑥} ∪ {𝑋}))
27	6, 7	lspss 18984	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ ({𝑥} ∪ {𝑋}) ⊆ 𝑉 ∧ {𝑥} ⊆ ({𝑥} ∪ {𝑋})) → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
28	3, 17, 26, 27	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
29		0ss 3972	. . . . . . . . . . . . . . 15 ⊢ ∅ ⊆ 𝑉
30	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ⊆ 𝑉)
31		uncom 3757	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∪ {𝑌}) = ({𝑌} ∪ ∅)
32		un0 3967	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑌} ∪ ∅) = {𝑌}
33	31, 32	eqtri 2644	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ∪ {𝑌}) = {𝑌}
34	33	fveq2i 6194	. . . . . . . . . . . . . . . 16 ⊢ (𝑁‘(∅ ∪ {𝑌})) = (𝑁‘{𝑌})
35	10, 34	syl6eleqr 2712	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑥 ∈ (𝑁‘(∅ ∪ {𝑌})))
36		lsppratlem1.x2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 }))
37	36	eldifbd 3587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑥 ∈ { 0 })
38		lsppratlem1.o	. . . . . . . . . . . . . . . . . 18 ⊢ 0 = (0_g‘𝑊)
39	38, 7	lsp0 19009	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
40	3, 39	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁‘∅) = { 0 })
41	37, 40	neleqtrrd 2723	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑥 ∈ (𝑁‘∅))
42	35, 41	eldifd 3585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))
43	6, 18, 7	lspsolv 19143	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LVec ∧ (∅ ⊆ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑥 ∈ ((𝑁‘(∅ ∪ {𝑌})) ∖ (𝑁‘∅)))) → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
44	1, 30, 4, 42, 43	syl13anc 1328	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘(∅ ∪ {𝑥})))
45		uncom 3757	. . . . . . . . . . . . . . 15 ⊢ (∅ ∪ {𝑥}) = ({𝑥} ∪ ∅)
46		un0 3967	. . . . . . . . . . . . . . 15 ⊢ ({𝑥} ∪ ∅) = {𝑥}
47	45, 46	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ (∅ ∪ {𝑥}) = {𝑥}
48	47	fveq2i 6194	. . . . . . . . . . . . 13 ⊢ (𝑁‘(∅ ∪ {𝑥})) = (𝑁‘{𝑥})
49	44, 48	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥}))
50	28, 49	sseldd 3604	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘({𝑥} ∪ {𝑋})))
51	50	snssd 4340	. . . . . . . . . 10 ⊢ (𝜑 → {𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
52	24, 51	unssd 3789	. . . . . . . . 9 ⊢ (𝜑 → ({𝑋} ∪ {𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
53	21, 52	syl5eqss 3649	. . . . . . . 8 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
54	18, 7	lspssp 18988	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ (𝑁‘({𝑥} ∪ {𝑋})) ∈ 𝑆 ∧ {𝑋, 𝑌} ⊆ (𝑁‘({𝑥} ∪ {𝑋}))) → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
55	3, 20, 53, 54	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
56	15, 55	sstrd 3613	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ (𝑁‘({𝑥} ∪ {𝑋})))
57	56	ssdifd 3746	. . . . 5 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
58		lsppratlem1.y2	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))
59	57, 58	sseldd 3604	. . . 4 ⊢ (𝜑 → 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))
60	6, 18, 7	lspsolv 19143	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ ({𝑥} ⊆ 𝑉 ∧ 𝑋 ∈ 𝑉 ∧ 𝑦 ∈ ((𝑁‘({𝑥} ∪ {𝑋})) ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
61	1, 12, 13, 59, 60	syl13anc 1328	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘({𝑥} ∪ {𝑦})))
62		df-pr 4180	. . . 4 ⊢ {𝑥, 𝑦} = ({𝑥} ∪ {𝑦})
63	62	fveq2i 6194	. . 3 ⊢ (𝑁‘{𝑥, 𝑦}) = (𝑁‘({𝑥} ∪ {𝑦}))
64	61, 63	syl6eleqr 2712	. 2 ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑥, 𝑦}))
65		lspprat.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
66	6, 18	lssss 18937	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉)
67	65, 66	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ 𝑉)
68	67	ssdifssd 3748	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∖ (𝑁‘{𝑥})) ⊆ 𝑉)
69	68, 58	sseldd 3604	. . . . . . 7 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
70	69	snssd 4340	. . . . . 6 ⊢ (𝜑 → {𝑦} ⊆ 𝑉)
71	12, 70	unssd 3789	. . . . 5 ⊢ (𝜑 → ({𝑥} ∪ {𝑦}) ⊆ 𝑉)
72	62, 71	syl5eqss 3649	. . . 4 ⊢ (𝜑 → {𝑥, 𝑦} ⊆ 𝑉)
73		snsspr1 4345	. . . . 5 ⊢ {𝑥} ⊆ {𝑥, 𝑦}
74	73	a1i 11	. . . 4 ⊢ (𝜑 → {𝑥} ⊆ {𝑥, 𝑦})
75	6, 7	lspss 18984	. . . 4 ⊢ ((𝑊 ∈ LMod ∧ {𝑥, 𝑦} ⊆ 𝑉 ∧ {𝑥} ⊆ {𝑥, 𝑦}) → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
76	3, 72, 74, 75	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝑁‘{𝑥}) ⊆ (𝑁‘{𝑥, 𝑦}))
77	76, 49	sseldd 3604	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))
78	64, 77	jca 554	1 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ⊊ wpss 3575 ∅c0 3915 {csn 4177 {cpr 4179 ‘cfv 5888 Basecbs 15857 0_gc0g 16100 LModclmod 18863 LSubSpclss 18932 LSpanclspn 18971 LVecclvec 19102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103

This theorem is referenced by: lsppratlem5 19151

W3C validator