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Theorem lspabs2 21122

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 21105	. . . . . . 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 20978	. . . . . 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 3963	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	5, 6	lspsnsubg 20978	. . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
12	3, 10, 11	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (SubGrp‘𝑊))
13		eqid 2737	. . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊)
14	13	lsmub2 19676	. . . . 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 19681	. . . . . 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 21094	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = (𝑁‘{𝑋, 𝑌}))
22	5, 20	lmodvacl 20873	. . . . . . . 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 21088	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, (𝑋 + 𝑌)}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})))
25	5, 6, 13, 3, 4, 10	lsmpr 21088	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
26	21, 24, 25	3eqtr3d 2785	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{(𝑋 + 𝑌)})) = ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})))
27	17, 19, 26	3eqtr3rd 2786	. . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑊)(𝑁‘{𝑌})) = (𝑁‘{𝑋}))
28	15, 27	sseqtrd 4020	. . 3 ⊢ (𝜑 → (𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}))
29		lspabs2.o	. . . 4 ⊢ 0 = (0_g‘𝑊)
30	5, 29, 6, 1, 9, 4	lspsncmp 21118	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑌}) ⊆ (𝑁‘{𝑋}) ↔ (𝑁‘{𝑌}) = (𝑁‘{𝑋})))
31	28, 30	mpbid 232	. 2 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑋}))
32	31	eqcomd 2743	1 ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 {cpr 4628 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +_gcplusg 17297 0_gc0g 17484 SubGrpcsubg 19138 LSSumclsm 19652 LModclmod 20858 LSpanclspn 20969 LVecclvec 21101

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102

This theorem is referenced by: lspindp3 21138

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