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Theorem lssssr 20215

Description: Conclude subspace ordering from nonzero vector membership. (ssrdv 3927 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 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 = (0_g‘𝑊)
5		lssssr.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑊)
6	4, 5	lss0cl 20208	. . . . . . 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 2839	. . . 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 3920	. . . . . 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 4720	. . . . . . . 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 3032	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈))
23	22	ssrdv 3927	1 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 ‘cfv 6433 0_gc0g 17150 LModclmod 20123 LSubSpclss 20193

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mgp 19721 df-ur 19738 df-ring 19785 df-lmod 20125 df-lss 20194

This theorem is referenced by: dihjat1lem 39442

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