dihjatcclem2 - Mathbox for Norm Megill

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Theorem dihjatcclem2 40881

Description: Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)

**Hypotheses**
Ref	Expression
dihjatcclem.b	⊢ 𝐵 = (Base‘𝐾)
dihjatcclem.l	⊢ ≤ = (le‘𝐾)
dihjatcclem.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjatcclem.j	⊢ ∨ = (join‘𝐾)
dihjatcclem.m	⊢ ∧ = (meet‘𝐾)
dihjatcclem.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjatcclem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjatcclem.s	⊢ ⊕ = (LSSum‘𝑈)
dihjatcclem.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihjatcclem.v	⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
dihjatcclem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihjatcclem.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dihjatcclem.q	⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
dihjatcclem2.c	⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

**Assertion**
Ref	Expression
dihjatcclem2	⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

**Proof of Theorem dihjatcclem2**
Step	Hyp	Ref	Expression
1		dihjatcclem.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dihjatcclem.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dihjatcclem.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dihjatcclem.j	. . 3 ⊢ ∨ = (join‘𝐾)
5		dihjatcclem.m	. . 3 ⊢ ∧ = (meet‘𝐾)
6		dihjatcclem.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
7		dihjatcclem.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		dihjatcclem.s	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
9		dihjatcclem.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
10		dihjatcclem.v	. . 3 ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
11		dihjatcclem.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		dihjatcclem.p	. . 3 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
13		dihjatcclem.q	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	dihjatcclem1 40880	. 2 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
15	3, 7, 11	dvhlmod 40572	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
16		eqid 2727	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
17	16	lsssssubg 20835	. . . . 5 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
18	15, 17	syl 17	. . . 4 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
19	12	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
20	1, 6	atbase 38750	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
21	19, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
22	1, 3, 9, 7, 16	dihlss 40712	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐵) → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
23	11, 21, 22	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑃) ∈ (LSubSp‘𝑈))
24	13	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
25	1, 6	atbase 38750	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
27	1, 3, 9, 7, 16	dihlss 40712	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
28	11, 26, 27	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑄) ∈ (LSubSp‘𝑈))
29	16, 8	lsmcl 20961	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘𝑃) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑄) ∈ (LSubSp‘𝑈)) → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (LSubSp‘𝑈))
30	15, 23, 28, 29	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (LSubSp‘𝑈))
31	18, 30	sseldd 3979	. . 3 ⊢ (𝜑 → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (SubGrp‘𝑈))
32	10	fveq2i 6894	. . . . 5 ⊢ (𝐼‘𝑉) = (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊))
33	11	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ HL)
34	33	hllatd 38825	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
35	1, 4, 6	hlatjcl 38828	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
36	33, 19, 24, 35	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ 𝐵)
37	11	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
38	1, 3	lhpbase 39460	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
39	37, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
40	1, 5	latmcl 18425	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
41	34, 36, 39, 40	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵)
42	1, 3, 9, 7, 16	dihlss 40712	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∧ 𝑊) ∈ 𝐵) → (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
43	11, 41, 42	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐼‘((𝑃 ∨ 𝑄) ∧ 𝑊)) ∈ (LSubSp‘𝑈))
44	32, 43	eqeltrid 2832	. . . 4 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (LSubSp‘𝑈))
45	18, 44	sseldd 3979	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ∈ (SubGrp‘𝑈))
46		dihjatcclem2.c	. . 3 ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
47	8	lsmss2 19615	. . 3 ⊢ ((((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ∈ (SubGrp‘𝑈) ∧ (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
48	31, 45, 46, 47	syl3anc 1369	. 2 ⊢ (𝜑 → (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
49	14, 48	eqtrd 2767	1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 lecple 17233 joincjn 18296 meetcmee 18297 Latclat 18416 SubGrpcsubg 19068 LSSumclsm 19582 LModclmod 20736 LSubSpclss 20808 Atomscatm 38724 HLchlt 38811 LHypclh 39446 DVecHcdvh 40540 DIsoHcdih 40690

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38414

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-oposet 38637 df-ol 38639 df-oml 38640 df-covers 38727 df-ats 38728 df-atl 38759 df-cvlat 38783 df-hlat 38812 df-llines 38960 df-lplanes 38961 df-lvols 38962 df-lines 38963 df-psubsp 38965 df-pmap 38966 df-padd 39258 df-lhyp 39450 df-laut 39451 df-ldil 39566 df-ltrn 39567 df-trl 39621 df-tendo 40217 df-edring 40219 df-disoa 40491 df-dvech 40541 df-dib 40601 df-dic 40635 df-dih 40691

This theorem is referenced by: dihjatcc 40884

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