dia2dimlem13 - Mathbox for Norm Megill

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Theorem dia2dimlem13 41063

Description: Lemma for dia2dim 41064. Eliminate 𝑈 ≠ 𝑉 condition. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem12.l	⊢ ≤ = (le‘𝐾)
dia2dimlem12.j	⊢ ∨ = (join‘𝐾)
dia2dimlem12.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem12.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem12.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem12.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem12.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem12.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem12.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem12.pl	⊢ ⊕ = (LSSum‘𝑌)
dia2dimlem12.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem12.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem12.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem12.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem12.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))

**Assertion**
Ref	Expression
dia2dimlem13	⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

**Proof of Theorem dia2dimlem13**
Step	Hyp	Ref	Expression
1		oveq2 7377	. . . . . . 7 ⊢ (𝑈 = 𝑉 → (𝑈 ∨ 𝑈) = (𝑈 ∨ 𝑉))
2	1	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑈 ∨ 𝑈) = (𝑈 ∨ 𝑉))
3		dia2dimlem12.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	3	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
5		dia2dimlem12.u	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
6	5	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
7		dia2dimlem12.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
8		dia2dimlem12.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
9	7, 8	hlatjidm 39355	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴) → (𝑈 ∨ 𝑈) = 𝑈)
10	4, 6, 9	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑈) = 𝑈)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑈 ∨ 𝑈) = 𝑈)
12	2, 11	eqtr3d 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝑈 ∨ 𝑉) = 𝑈)
13	12	fveq2d 6844	. . . 4 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐼‘(𝑈 ∨ 𝑉)) = (𝐼‘𝑈))
14		ssid 3966	. . . 4 ⊢ (𝐼‘𝑈) ⊆ (𝐼‘𝑈)
15	13, 14	eqsstrdi 3988	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ (𝐼‘𝑈))
16		dia2dimlem12.h	. . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾)
17		dia2dimlem12.y	. . . . . . . 8 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
18	16, 17	dvalvec 41013	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑌 ∈ LVec)
19		lveclmod 21045	. . . . . . 7 ⊢ (𝑌 ∈ LVec → 𝑌 ∈ LMod)
20	3, 18, 19	3syl 18	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ LMod)
21		eqid 2729	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 8	atbase 39275	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
23	6, 22	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
24	5	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
25		dia2dimlem12.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
26		dia2dimlem12.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
27		dia2dimlem12.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑌)
28	21, 25, 16, 17, 26, 27	dialss 41033	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ (Base‘𝐾) ∧ 𝑈 ≤ 𝑊)) → (𝐼‘𝑈) ∈ 𝑆)
29	3, 23, 24, 28	syl12anc 836	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑈) ∈ 𝑆)
30	27	lsssubg 20895	. . . . . 6 ⊢ ((𝑌 ∈ LMod ∧ (𝐼‘𝑈) ∈ 𝑆) → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
31	20, 29, 30	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑈) ∈ (SubGrp‘𝑌))
32		dia2dimlem12.pl	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑌)
33	32	lsmidm 19577	. . . . 5 ⊢ ((𝐼‘𝑈) ∈ (SubGrp‘𝑌) → ((𝐼‘𝑈) ⊕ (𝐼‘𝑈)) = (𝐼‘𝑈))
34	31, 33	syl 17	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑈) ⊕ (𝐼‘𝑈)) = (𝐼‘𝑈))
35		fveq2 6840	. . . . 5 ⊢ (𝑈 = 𝑉 → (𝐼‘𝑈) = (𝐼‘𝑉))
36	35	oveq2d 7385	. . . 4 ⊢ (𝑈 = 𝑉 → ((𝐼‘𝑈) ⊕ (𝐼‘𝑈)) = ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
37	34, 36	sylan9req 2785	. . 3 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐼‘𝑈) = ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
38	15, 37	sseqtrd 3980	. 2 ⊢ ((𝜑 ∧ 𝑈 = 𝑉) → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
39		dia2dimlem12.m	. . 3 ⊢ ∧ = (meet‘𝐾)
40		dia2dimlem12.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
41		dia2dimlem12.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
42		dia2dimlem12.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑌)
43	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ 𝑉) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
44	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ 𝑉) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
45		dia2dimlem12.v	. . . 4 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
46	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ 𝑉) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
47		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑈 ≠ 𝑉) → 𝑈 ≠ 𝑉)
48	25, 7, 39, 8, 16, 40, 41, 17, 27, 32, 42, 26, 43, 44, 46, 47	dia2dimlem12 41062	. 2 ⊢ ((𝜑 ∧ 𝑈 ≠ 𝑉) → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
49	38, 48	pm2.61dane 3012	1 ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3911 class class class wbr 5102 ‘cfv 6499 (class class class)co 7369 Basecbs 17155 lecple 17203 joincjn 18252 meetcmee 18253 SubGrpcsubg 19034 LSSumclsm 19548 LModclmod 20798 LSubSpclss 20869 LSpanclspn 20909 LVecclvec 21041 Atomscatm 39249 HLchlt 39336 LHypclh 39971 LTrncltrn 40088 trLctrl 40145 DVecAcdveca 40989 DIsoAcdia 41015

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-riotaBAD 38939

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-undef 8229 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17380 df-proset 18235 df-poset 18254 df-plt 18269 df-lub 18285 df-glb 18286 df-join 18287 df-meet 18288 df-p0 18364 df-p1 18365 df-lat 18373 df-clat 18440 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-subg 19037 df-cntz 19231 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-lvec 21042 df-oposet 39162 df-ol 39164 df-oml 39165 df-covers 39252 df-ats 39253 df-atl 39284 df-cvlat 39308 df-hlat 39337 df-llines 39485 df-lplanes 39486 df-lvols 39487 df-lines 39488 df-psubsp 39490 df-pmap 39491 df-padd 39783 df-lhyp 39975 df-laut 39976 df-ldil 40091 df-ltrn 40092 df-trl 40146 df-tgrp 40730 df-tendo 40742 df-edring 40744 df-dveca 40990 df-disoa 41016

This theorem is referenced by: dia2dim 41064

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