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Theorem lspabs2 21170

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 21153	. . . . . . 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 21027	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
8	3, 4, 7	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
9		lspabs2.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
10	9	eldifad 3916	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	5, 6	lspsnsubg 21027	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
12	3, 10, 11	syl2anc 593	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
13		eqid 2761	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	13	lsmub2 19681	. . . . 5 ⊢ (((𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊)) → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
15	8, 12, 14	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
16		lspabs2.e	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{(𝑋 + 𝑌)}))
17	16	oveq2d 7408	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑋})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
18	13	lsmidm 19686	. . . . . 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 21142	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
22	5, 20	lmodvacl 20922	. . . . . . . 8 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 + 𝑌) ∈ 𝑉)
23	3, 4, 10, 22	syl3anc 1389	. . . . . . 7 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
24	5, 6, 13, 3, 4, 23	lsmpr 21136	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
25	5, 6, 13, 3, 4, 10	lsmpr 21136	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
26	21, 24, 25	3eqtr3d 2804	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
27	17, 19, 26	3eqtr3rd 2805	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋}))
28	15, 27	sseqtrd 3972	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}))
29		lspabs2.o	. . . 4 ⊢ 0 = (0_g‘𝑊)
30	5, 29, 6, 1, 9, 4	lspsncmp 21166	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋})))
31	28, 30	mpbid 234	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
32	31	eqcomd 2767	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ∖ cdif 3901 ⊆ wss 3904 {csn 4581 {cpr 4583 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 0_gc0g 17451 SubGrpcsubg 19145 LSSumclsm 19657 LModclmod 20907 LSpanclspn 21018 LVecclvec 21149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-lvec 21150

This theorem is referenced by: lspindp3 21186

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