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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 3984 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 485 | . . . . 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 20506 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) |
7 | 2, 3, 6 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝑈) |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 0 ∈ 𝑈) |
9 | 1, 8 | eqeltrd 2832 | . . . 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 3977 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑉)) |
13 | 12 | ancrd 552 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇))) |
14 | 13 | adantr 481 | . . . 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 595 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
18 | 17 | exp32 421 | . . . . . 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 422 | . . . 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 3028 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
23 | 22 | ssrdv 3984 | 1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3941 ⊆ wss 3944 {csn 4622 ‘cfv 6532 0gc0g 17367 LModclmod 20420 LSubSpclss 20491 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-0g 17369 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-sbg 18799 df-mgp 19947 df-ur 19964 df-ring 20016 df-lmod 20422 df-lss 20492 |
This theorem is referenced by: dihjat1lem 40104 |
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