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Theorem lspabs2 21045

Description: Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015.)

**Hypotheses**
Ref	Expression
lspabs2.v	⊢ 𝑉 = (Base‘𝑊)
lspabs2.p	⊢ + = (+_g‘𝑊)
lspabs2.o	⊢ 0 = (0_g‘𝑊)
lspabs2.n	⊢ 𝑁 = (LSpan‘𝑊)
lspabs2.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lspabs2.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lspabs2.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lspabs2.e	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))

**Assertion**
Ref	Expression
lspabs2	⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

**Proof of Theorem lspabs2**
Step	Hyp	Ref	Expression
1		lspabs2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec)
2		lveclmod 21028	. . . . . . 7 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ LMod)
4		lspabs2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
5		lspabs2.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
6		lspabs2.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑊)
7	5, 6	lspsnsubg 20901	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
8	3, 4, 7	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
9		lspabs2.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3917	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	5, 6	lspsnsubg 20901	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
12	3, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
13		eqid 2729	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	13	lsmub2 19555	. . . . 5 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
15	8, 12, 14	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
16		lspabs2.e	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))
17	16	oveq2d 7369	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
18	13	lsmidm 19560	. . . . . 6 ⊢ ((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋}))
19	8, 18	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = (𝑁‘{𝑋}))
20		lspabs2.p	. . . . . . 7 ⊢ + = (+_g‘𝑊)
21	5, 20, 6, 3, 4, 10	lspprabs 21017	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
22	5, 20	lmodvacl 20796	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
23	3, 4, 10, 22	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
24	5, 6, 13, 3, 4, 23	lsmpr 21011	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
25	5, 6, 13, 3, 4, 10	lsmpr 21011	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
26	21, 24, 25	3eqtr3d 2772	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
27	17, 19, 26	3eqtr3rd 2773	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋}))
28	15, 27	sseqtrd 3974	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}))
29		lspabs2.o	. . . 4 ⊢ 0 = (0_g‘𝑊)
30	5, 29, 6, 1, 9, 4	lspsncmp 21041	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋})))
31	28, 30	mpbid 232	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
32	31	eqcomd 2735	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∖ cdif 3902 ⊆ wss 3905 {csn 4579 {cpr 4581 ‘cfv 6486 (class class class)co 7353 Basecbs 17138 +_gcplusg 17179 0_gc0g 17361 SubGrpcsubg 19017 LSSumclsm 19531 LModclmod 20781 LSpanclspn 20892 LVecclvec 21024

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-0g 17363 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lvec 21025

This theorem is referenced by: lspindp3 21061

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