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Mirrors > Home > MPE Home > Th. List > lsppratlem6 | Structured version Visualization version GIF version |
Description: Lemma for lspprat 19367. 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 466 | . . . . . 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 466 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑊 ∈ LVec) |
8 | lspprat.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
9 | 8 | adantr 466 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ∈ 𝑆) |
10 | lspprat.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | 10 | adantr 466 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑋 ∈ 𝑉) |
12 | lspprat.y | . . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
13 | 12 | adantr 466 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑌 ∈ 𝑉) |
14 | 1 | adantr 466 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑈 ⊊ (𝑁‘{𝑋, 𝑌})) |
15 | lsppratlem6.o | . . . . . . . . 9 ⊢ 0 = (0g‘𝑊) | |
16 | simprl 746 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → 𝑥 ∈ (𝑈 ∖ { 0 })) | |
17 | simprr 748 | . . . . . . . . 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 19365 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑈 ∖ { 0 }) ∧ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})))) → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) |
19 | ssnpss 3858 | . . . . . . . 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 444 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥})) → ¬ 𝑈 ⊊ (𝑁‘{𝑋, 𝑌}))) |
22 | 2, 21 | mt2d 133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → ¬ 𝑦 ∈ (𝑈 ∖ (𝑁‘{𝑥}))) |
23 | 22 | eq0rdv 4121 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑈 ∖ (𝑁‘{𝑥})) = ∅) |
24 | ssdif0 4087 | . . . 4 ⊢ (𝑈 ⊆ (𝑁‘{𝑥}) ↔ (𝑈 ∖ (𝑁‘{𝑥})) = ∅) | |
25 | 23, 24 | sylibr 224 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ⊆ (𝑁‘{𝑥})) |
26 | lveclmod 19318 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
27 | 6, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
28 | 27 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑊 ∈ LMod) |
29 | 8 | adantr 466 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 ∈ 𝑆) |
30 | eldifi 3881 | . . . . 5 ⊢ (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑥 ∈ 𝑈) | |
31 | 30 | adantl 467 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑥 ∈ 𝑈) |
32 | 4, 5, 28, 29, 31 | lspsnel5a 19208 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → (𝑁‘{𝑥}) ⊆ 𝑈) |
33 | 25, 32 | eqssd 3767 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑈 ∖ { 0 })) → 𝑈 = (𝑁‘{𝑥})) |
34 | 33 | ex 397 | 1 ⊢ (𝜑 → (𝑥 ∈ (𝑈 ∖ { 0 }) → 𝑈 = (𝑁‘{𝑥}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 ∖ cdif 3718 ⊆ wss 3721 ⊊ wpss 3722 ∅c0 4061 {csn 4314 {cpr 4316 ‘cfv 6031 Basecbs 16063 0gc0g 16307 LModclmod 19072 LSubSpclss 19141 LSpanclspn 19183 LVecclvec 19314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-ndx 16066 df-slot 16067 df-base 16069 df-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-0g 16309 df-mgm 17449 df-sgrp 17491 df-mnd 17502 df-grp 17632 df-minusg 17633 df-sbg 17634 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 |
This theorem is referenced by: lspprat 19367 |
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