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Theorem lspabs2 21140

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 21123	. . . . . . 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 20996	. . . . . 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 3975	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	5, 6	lspsnsubg 20996	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
12	3, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
13		eqid 2735	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	13	lsmub2 19691	. . . . 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 7447	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
18	13	lsmidm 19696	. . . . . 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 21112	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
22	5, 20	lmodvacl 20890	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
23	3, 4, 10, 22	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
24	5, 6, 13, 3, 4, 23	lsmpr 21106	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
25	5, 6, 13, 3, 4, 10	lsmpr 21106	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
26	21, 24, 25	3eqtr3d 2783	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
27	17, 19, 26	3eqtr3rd 2784	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋}))
28	15, 27	sseqtrd 4036	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}))
29		lspabs2.o	. . . 4 ⊢ 0 = (0_g‘𝑊)
30	5, 29, 6, 1, 9, 4	lspsncmp 21136	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋})))
31	28, 30	mpbid 232	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
32	31	eqcomd 2741	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 0_gc0g 17486 SubGrpcsubg 19151 LSSumclsm 19667 LModclmod 20875 LSpanclspn 20987 LVecclvec 21119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120

This theorem is referenced by: lspindp3 21156

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