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Mirrors > Home > MPE Home > Th. List > lsppratlem6 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 19854. Negating the assumption on 𝑦, we arrive close to the desired conclusion. (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})) |
lsppratlem6.o | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
lsppratlem6 | ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspprat.p | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
2 | 1 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
3 | lspprat.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
4 | lspprat.s | . . . . . . . . 9 ⊢ 𝑆 = (LSubSp‘𝑊) | |
5 | lspprat.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | lspprat.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | 6 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑊 ∈ LVec) |
8 | lspprat.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
9 | 8 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ∈ 𝑆) |
10 | lspprat.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ 𝑉) |
12 | lspprat.y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | 12 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑌 ∈ 𝑉) |
14 | 1 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
15 | lsppratlem6.o | . . . . . . . . 9 ⊢ 0 = (0g‘𝑊) | |
16 | simprl 767 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
17 | simprr 769 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) | |
18 | 3, 4, 5, 7, 9, 11, 13, 14, 15, 16, 17 | lsppratlem5 19852 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
19 | ssnpss 4077 | . . . . . . . 8 ⊢ ((𝑁‘{𝑋, 𝑌}) ⊆ 𝑈 → ¬ 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) | |
20 | 18, 19 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → ¬ 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
21 | 20 | expr 457 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) → ¬ 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))) |
22 | 2, 21 | mt2d 138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → ¬ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
23 | 22 | eq0rdv 4354 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑈 ∖ (𝑁‘{𝑥})) = ∅) |
24 | ssdif0 4320 | . . . 4 ⊢ (𝑈 ⊆ (𝑁‘{𝑥}) ↔ (𝑈 ∖ (𝑁‘{𝑥})) = ∅) | |
25 | 23, 24 | sylibr 235 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ⊆ (𝑁‘{𝑥})) |
26 | lveclmod 19807 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
27 | 6, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
28 | 27 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑊 ∈ LMod) |
29 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ∈ 𝑆) |
30 | eldifi 4100 | . . . . 5 ⊢ (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑥 ∈ 𝑈) | |
31 | 30 | adantl 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑥 ∈ 𝑈) |
32 | 4, 5, 28, 29, 31 | lspsnel5a 19697 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑁‘{𝑥}) ⊆ 𝑈) |
33 | 25, 32 | eqssd 3981 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 = (𝑁‘{𝑥})) |
34 | 33 | ex 413 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ⊆ wss 3933 ⊊ wpss 3934 ∅c0 4288 {csn 4557 {cpr 4559 ‘cfv 6348 Basecbs 16471 0gc0g 16701 LModclmod 19563 LSubSpclss 19632 LSpanclspn 19672 LVecclvec 19803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 |
This theorem is referenced by: lspprat 19854 |
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