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Mirrors > Home > MPE Home > Th. List > lsppratlem5 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 21030. 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑊 ∈ LVec) |
6 | lspprat.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑈 ∈ 𝑆) |
8 | lspprat.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑋 ∈ 𝑉) |
10 | lspprat.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑌 ∈ 𝑉) |
12 | lspprat.p | . . . . 5 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
13 | 12 | adantr 480 | . . . 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
17 | lsppratlem1.y2 | . . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) | |
18 | 17 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
19 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → 𝑥 ∈ (𝑁‘{𝑌})) | |
20 | 1, 2, 3, 5, 7, 9, 11, 13, 14, 16, 18, 19 | lsppratlem3 21026 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑁‘{𝑌})) → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) |
21 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑊 ∈ LVec) |
22 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑈 ∈ 𝑆) |
23 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑋 ∈ 𝑉) |
24 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑌 ∈ 𝑉) |
25 | 12 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
26 | 15 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
27 | 17 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
28 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) → 𝑋 ∈ (𝑁‘{𝑥, 𝑌})) | |
29 | 1, 2, 3, 21, 22, 23, 24, 25, 14, 26, 27, 28 | lsppratlem4 21027 | . . 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 21024 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘{𝑌}) ∨ 𝑋 ∈ (𝑁‘{𝑥, 𝑌}))) |
31 | 20, 29, 30 | mpjaodan 957 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) |
32 | 4 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑊 ∈ LVec) |
33 | 6 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑈 ∈ 𝑆) |
34 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑋 ∈ 𝑉) |
35 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑌 ∈ 𝑉) |
36 | 12 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
37 | 15 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑥 ∈ (𝑈 ∖ { 0 })) |
38 | 17 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
39 | simprl 770 | . . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → 𝑋 ∈ (𝑁‘{𝑥, 𝑦})) | |
40 | simprr 772 | . . 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 21025 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ (𝑁‘{𝑥, 𝑦}) ∧ 𝑌 ∈ (𝑁‘{𝑥, 𝑦}))) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
42 | 31, 41 | mpdan 686 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∖ cdif 3941 ⊆ wss 3944 ⊊ wpss 3945 {csn 4624 {cpr 4626 ‘cfv 6542 Basecbs 17171 0gc0g 17412 LSubSpclss 20804 LSpanclspn 20844 LVecclvec 20976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-sbg 18886 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-drng 20615 df-lmod 20734 df-lss 20805 df-lsp 20845 df-lvec 20977 |
This theorem is referenced by: lsppratlem6 21029 |
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