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Mirrors > Home > MPE Home > Th. List > lsppratlem5 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 19521. Combine the two cases and show a contradiction to 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}) under the assumptions on 𝑥 and 𝑦. (Contributed by NM, 29-Aug-2014.) |
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 = (0g‘𝑊) |
lsppratlem1.x2 | ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) |
lsppratlem1.y2 | ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
Ref | Expression |
---|---|
lsppratlem5 | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprat.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | lspprat.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
3 | lspprat.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑊) | |
4 | lspprat.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
6 | lspprat.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑈 ∈ 𝑆) |
8 | lspprat.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ 𝑉) |
10 | lspprat.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑌 ∈ 𝑉) |
12 | lspprat.p | . . . . 5 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
13 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
14 | lsppratlem1.o | . . . 4 ⊢ 0 = (0g‘𝑊) | |
15 | lsppratlem1.x2 | . . . . 5 ⊢ (𝜑 → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
16 | 15 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
17 | lsppratlem1.y2 | . . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) | |
18 | 17 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
19 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑥 ∈ (𝑁‘{𝑌})) | |
20 | 1, 2, 3, 5, 7, 9, 11, 13, 14, 16, 18, 19 | lsppratlem3 19517 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) |
21 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑊 ∈ LVec) |
22 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑈 ∈ 𝑆) |
23 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑋 ∈ 𝑉) |
24 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑌 ∈ 𝑉) |
25 | 12 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
26 | 15 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
27 | 17 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
28 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) | |
29 | 1, 2, 3, 21, 22, 23, 24, 25, 14, 26, 27, 28 | lsppratlem4 19518 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) |
30 | 1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 17 | lsppratlem1 19515 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
31 | 20, 29, 30 | mpjaodan 986 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) |
32 | 4 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑊 ∈ LVec) |
33 | 6 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑈 ∈ 𝑆) |
34 | 8 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑋 ∈ 𝑉) |
35 | 10 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑌 ∈ 𝑉) |
36 | 12 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
37 | 15 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
38 | 17 | adantr 474 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
39 | simprl 787 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) | |
40 | simprr 789 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑌 ∈ (𝑁‘{𝑥, 𝑦})) | |
41 | 1, 2, 3, 32, 33, 34, 35, 36, 14, 37, 38, 39, 40 | lsppratlem2 19516 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
42 | 31, 41 | mpdan 678 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∖ cdif 3795 ⊆ wss 3798 ⊊ wpss 3799 {csn 4399 {cpr 4401 ‘cfv 6127 Basecbs 16229 0gc0g 16460 LSubSpclss 19295 LSpanclspn 19337 LVecclvec 19468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 |
This theorem is referenced by: lsppratlem6 19520 |
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