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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lssatle | Structured version Visualization version GIF version |
Description: The ordering of two subspaces is determined by the atoms under them. (chrelat3 29755 analog.) (Contributed by NM, 29-Oct-2014.) |
Ref | Expression |
---|---|
lssatle.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssatle.a | ⊢ 𝐴 = (LSAtoms‘𝑊) |
lssatle.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssatle.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
lssatle.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
Ref | Expression |
---|---|
lssatle | ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr 3806 | . . . 4 ⊢ ((𝑝 ⊆ 𝑇 ∧ 𝑇 ⊆ 𝑈) → 𝑝 ⊆ 𝑈) | |
2 | 1 | expcom 403 | . . 3 ⊢ (𝑇 ⊆ 𝑈 → (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈)) |
3 | 2 | ralrimivw 3148 | . 2 ⊢ (𝑇 ⊆ 𝑈 → ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈)) |
4 | ss2rab 3874 | . . 3 ⊢ ({𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈)) | |
5 | lssatle.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
6 | 5 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) → 𝑊 ∈ LMod) |
7 | lssatle.s | . . . . . . . . . 10 ⊢ 𝑆 = (LSubSp‘𝑊) | |
8 | lssatle.a | . . . . . . . . . 10 ⊢ 𝐴 = (LSAtoms‘𝑊) | |
9 | 7, 8 | lsatlss 35017 | . . . . . . . . 9 ⊢ (𝑊 ∈ LMod → 𝐴 ⊆ 𝑆) |
10 | rabss2 3881 | . . . . . . . . 9 ⊢ (𝐴 ⊆ 𝑆 → {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈}) | |
11 | uniss 4651 | . . . . . . . . 9 ⊢ ({𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈} → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ ∪ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈}) | |
12 | 5, 9, 10, 11 | 4syl 19 | . . . . . . . 8 ⊢ (𝜑 → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ ∪ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈}) |
13 | lssatle.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
14 | unimax 4665 | . . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → ∪ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈} = 𝑈) | |
15 | 13, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → ∪ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈} = 𝑈) |
16 | eqid 2799 | . . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
17 | 16, 7 | lssss 19255 | . . . . . . . . . 10 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ (Base‘𝑊)) |
18 | 13, 17 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝑊)) |
19 | 15, 18 | eqsstrd 3835 | . . . . . . . 8 ⊢ (𝜑 → ∪ {𝑝 ∈ 𝑆 ∣ 𝑝 ⊆ 𝑈} ⊆ (Base‘𝑊)) |
20 | 12, 19 | sstrd 3808 | . . . . . . 7 ⊢ (𝜑 → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ (Base‘𝑊)) |
21 | 20 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ (Base‘𝑊)) |
22 | uniss 4651 | . . . . . . 7 ⊢ ({𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) | |
23 | 22 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) → ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) |
24 | eqid 2799 | . . . . . . 7 ⊢ (LSpan‘𝑊) = (LSpan‘𝑊) | |
25 | 16, 24 | lspss 19305 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} ⊆ (Base‘𝑊) ∧ ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ ∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) → ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇}) ⊆ ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈})) |
26 | 6, 21, 23, 25 | syl3anc 1491 | . . . . 5 ⊢ ((𝜑 ∧ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}) → ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇}) ⊆ ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈})) |
27 | 26 | ex 402 | . . . 4 ⊢ (𝜑 → ({𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} → ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇}) ⊆ ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}))) |
28 | lssatle.t | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
29 | 7, 24, 8 | lssats 35033 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆) → 𝑇 = ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇})) |
30 | 5, 28, 29 | syl2anc 580 | . . . . 5 ⊢ (𝜑 → 𝑇 = ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇})) |
31 | 7, 24, 8 | lssats 35033 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 = ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈})) |
32 | 5, 13, 31 | syl2anc 580 | . . . . 5 ⊢ (𝜑 → 𝑈 = ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈})) |
33 | 30, 32 | sseq12d 3830 | . . . 4 ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇}) ⊆ ((LSpan‘𝑊)‘∪ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈}))) |
34 | 27, 33 | sylibrd 251 | . . 3 ⊢ (𝜑 → ({𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑇} ⊆ {𝑝 ∈ 𝐴 ∣ 𝑝 ⊆ 𝑈} → 𝑇 ⊆ 𝑈)) |
35 | 4, 34 | syl5bir 235 | . 2 ⊢ (𝜑 → (∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈) → 𝑇 ⊆ 𝑈)) |
36 | 3, 35 | impbid2 218 | 1 ⊢ (𝜑 → (𝑇 ⊆ 𝑈 ↔ ∀𝑝 ∈ 𝐴 (𝑝 ⊆ 𝑇 → 𝑝 ⊆ 𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ⊆ wss 3769 ∪ cuni 4628 ‘cfv 6101 Basecbs 16184 LModclmod 19181 LSubSpclss 19250 LSpanclspn 19292 LSAtomsclsa 34995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-plusg 16280 df-0g 16417 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-grp 17741 df-minusg 17742 df-sbg 17743 df-mgp 18806 df-ur 18818 df-ring 18865 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lsatoms 34997 |
This theorem is referenced by: mapdordlem2 37658 |
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