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Mirrors > Home > MPE Home > Th. List > lsppratlem6 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 19918. 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 484 | . . . . . 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 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑊 ∈ LVec) |
8 | lspprat.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
9 | 8 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ∈ 𝑆) |
10 | lspprat.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ 𝑉) |
12 | lspprat.y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | 12 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑌 ∈ 𝑉) |
14 | 1 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
15 | lsppratlem6.o | . . . . . . . . 9 ⊢ 0 = (0g‘𝑊) | |
16 | simprl 770 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
17 | simprr 772 | . . . . . . . . 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 19916 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
19 | ssnpss 4031 | . . . . . . . 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 460 | . . . . . 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 4312 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑈 ∖ (𝑁‘{𝑥})) = ∅) |
24 | ssdif0 4277 | . . . 4 ⊢ (𝑈 ⊆ (𝑁‘{𝑥}) ↔ (𝑈 ∖ (𝑁‘{𝑥})) = ∅) | |
25 | 23, 24 | sylibr 237 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ⊆ (𝑁‘{𝑥})) |
26 | lveclmod 19871 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
27 | 6, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
28 | 27 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑊 ∈ LMod) |
29 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ∈ 𝑆) |
30 | eldifi 4054 | . . . . 5 ⊢ (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑥 ∈ 𝑈) | |
31 | 30 | adantl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑥 ∈ 𝑈) |
32 | 4, 5, 28, 29, 31 | lspsnel5a 19761 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑁‘{𝑥}) ⊆ 𝑈) |
33 | 25, 32 | eqssd 3932 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 = (𝑁‘{𝑥})) |
34 | 33 | ex 416 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ⊊ wpss 3882 ∅c0 4243 {csn 4525 {cpr 4527 ‘cfv 6324 Basecbs 16475 0gc0g 16705 LModclmod 19627 LSubSpclss 19696 LSpanclspn 19736 LVecclvec 19867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 |
This theorem is referenced by: lspprat 19918 |
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