lssssr - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > lssssr		Structured version Visualization version GIF version

Theorem lssssr 20564

Description: Conclude subspace ordering from nonzero vector membership. (ssrdv 3989 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.)

**Hypotheses**
Ref	Expression
lssssr.o	⊢ 0 = (0_g‘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))

**Assertion**
Ref	Expression
lssssr	⊢ (𝜑 → 𝑇 ⊆ 𝑈)

Distinct variable groups: 𝑥,𝑇 𝑥,𝑈 𝜑,𝑥 Allowed substitution hints: 𝑆(𝑥) 𝑉(𝑥) 𝑊(𝑥) 0 (𝑥)

**Proof of Theorem lssssr**
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 = (0_g‘𝑊)
5		lssssr.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	lss0cl 20557	. . . . . . 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 3982	. . . . . 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 4791	. . . . . . . 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 3030	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈))
23	22	ssrdv 3989	1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∖ cdif 3946 ⊆ wss 3949 {csn 4629 ‘cfv 6544 0_gc0g 17385 LModclmod 20471 LSubSpclss 20542

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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mgp 19988 df-ur 20005 df-ring 20058 df-lmod 20473 df-lss 20543

This theorem is referenced by: dihjat1lem 40299

W3C validator