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Mirrors > Home > MPE Home > Th. List > lssssr | Structured version Visualization version GIF version |
Description: Conclude subspace ordering from nonzero vector membership. (ssrdv 3981 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.) |
Ref | Expression |
---|---|
lssssr.o | ⊢ 0 = (0g‘𝑊) |
lssssr.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
lssssr.w | ⊢ (𝜑 → 𝑊 ∈ LMod) |
lssssr.t | ⊢ (𝜑 → 𝑇 ⊆ 𝑉) |
lssssr.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
lssssr.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
Ref | Expression |
---|---|
lssssr | ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 𝑥 = 0 ) | |
2 | lssssr.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
3 | lssssr.u | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
4 | lssssr.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
5 | lssssr.s | . . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊) | |
6 | 4, 5 | lss0cl 20790 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) |
7 | 2, 3, 6 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝑈) |
8 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 0 ∈ 𝑈) |
9 | 1, 8 | eqeltrd 2825 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 𝑥 ∈ 𝑈) |
10 | 9 | a1d 25 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
11 | lssssr.t | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ 𝑉) | |
12 | 11 | sseld 3974 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑉)) |
13 | 12 | ancrd 551 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇))) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ≠ 0 ) → (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇))) |
15 | eldifsn 4783 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) | |
16 | lssssr.1 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) | |
17 | 15, 16 | sylan2br 594 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
18 | 17 | exp32 420 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑉 → (𝑥 ≠ 0 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)))) |
19 | 18 | com23 86 | . . . . 5 ⊢ (𝜑 → (𝑥 ≠ 0 → (𝑥 ∈ 𝑉 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)))) |
20 | 19 | imp4b 421 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ≠ 0 ) → ((𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑈)) |
21 | 14, 20 | syld 47 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ≠ 0 ) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
22 | 10, 21 | pm2.61dane 3021 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
23 | 22 | ssrdv 3981 | 1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3938 ⊆ wss 3941 {csn 4621 ‘cfv 6534 0gc0g 17390 LModclmod 20702 LSubSpclss 20774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-0g 17392 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mgp 20036 df-ur 20083 df-ring 20136 df-lmod 20704 df-lss 20775 |
This theorem is referenced by: dihjat1lem 40803 |
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