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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 3951 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 486 | . . . . 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 20422 | . . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) |
7 | 2, 3, 6 | syl2anc 585 | . . . . . 6 ⊢ (𝜑 → 0 ∈ 𝑈) |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 0 ∈ 𝑈) |
9 | 1, 8 | eqeltrd 2834 | . . . 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 3944 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑉)) |
13 | 12 | ancrd 553 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇))) |
14 | 13 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ≠ 0 ) → (𝑥 ∈ 𝑇 → (𝑥 ∈ 𝑉 ∧ 𝑥 ∈ 𝑇))) |
15 | eldifsn 4748 | . . . . . . . 8 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) ↔ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) | |
16 | lssssr.1 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) | |
17 | 15, 16 | sylan2br 596 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
18 | 17 | exp32 422 | . . . . . 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 423 | . . . 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 3029 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) |
23 | 22 | ssrdv 3951 | 1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∖ cdif 3908 ⊆ wss 3911 {csn 4587 ‘cfv 6497 0gc0g 17326 LModclmod 20336 LSubSpclss 20407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-minusg 18757 df-sbg 18758 df-mgp 19902 df-ur 19919 df-ring 19971 df-lmod 20338 df-lss 20408 |
This theorem is referenced by: dihjat1lem 39937 |
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