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Theorem lssssr 20916

Description: Conclude subspace ordering from nonzero vector membership. (ssrdv 3969 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 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 = (0_g‘𝑊)
5		lssssr.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	lss0cl 20909	. . . . . . 7 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈)
7	2, 3, 6	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 0 ∈ 𝑈)
8	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0 ) → 0 ∈ 𝑈)
9	1, 8	eqeltrd 2835	. . . 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 3962	. . . . . 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 4767	. . . . . . . 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 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 3020	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈))
23	22	ssrdv 3969	1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∖ cdif 3928 ⊆ wss 3931 {csn 4606 ‘cfv 6536 0_gc0g 17458 LModclmod 20822 LSubSpclss 20893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-plusg 17289 df-0g 17460 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mgp 20106 df-ur 20147 df-ring 20200 df-lmod 20824 df-lss 20894

This theorem is referenced by: dihjat1lem 41452

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