dihordlem6 - Mathbox for Norm Megill

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Theorem dihordlem6 41670

Description: Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)

**Hypotheses**
Ref	Expression
dihordlem8.b	⊢ 𝐵 = (Base‘𝐾)
dihordlem8.l	⊢ ≤ = (le‘𝐾)
dihordlem8.a	⊢ 𝐴 = (Atoms‘𝐾)
dihordlem8.h	⊢ 𝐻 = (LHyp‘𝐾)
dihordlem8.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihordlem8.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihordlem8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihordlem8.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihordlem8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihordlem8.s	⊢ + = (+_g‘𝑈)
dihordlem8.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)

**Assertion**
Ref	Expression
dihordlem6	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)

Distinct variable groups: ≤ ,ℎ 𝐴,ℎ 𝐵,ℎ ℎ,𝐻 ℎ,𝐾 𝑃,ℎ 𝑅,ℎ 𝑇,ℎ ℎ,𝑊 Allowed substitution hints: 𝐴(𝑔,𝑠) 𝐵(𝑔,𝑠) 𝑃(𝑔,𝑠) + (𝑔,ℎ,𝑠) 𝑄(𝑔,ℎ,𝑠) 𝑅(𝑔,𝑠) 𝑇(𝑔,𝑠) 𝑈(𝑔,ℎ,𝑠) 𝐸(𝑔,ℎ,𝑠) 𝐺(𝑔,ℎ,𝑠) 𝐻(𝑔,𝑠) 𝐾(𝑔,𝑠) ≤ (𝑔,𝑠) 𝑂(𝑔,ℎ,𝑠) 𝑊(𝑔,𝑠)

**Proof of Theorem dihordlem6**
Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simp2r 1202	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
3		simp2l 1201	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
4		simp3 1139	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇))
5		dihordlem8.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
6		dihordlem8.l	. . 3 ⊢ ≤ = (le‘𝐾)
7		dihordlem8.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
8		dihordlem8.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
9		dihordlem8.p	. . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
10		dihordlem8.o	. . 3 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
11		dihordlem8.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
12		dihordlem8.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
13		dihordlem8.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
14		dihordlem8.s	. . 3 ⊢ + = (+_g‘𝑈)
15		dihordlem8.g	. . 3 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
16	5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	cdlemn6 41659	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)
17	1, 2, 3, 4, 16	syl121anc 1378	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) = ⟨((𝑠‘𝐺) ∘ 𝑔), 𝑠⟩)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ⟨cop 4574 class class class wbr 5086 ↦ cmpt 5167 I cid 5516 ↾ cres 5624 ∘ ccom 5626 ‘cfv 6490 ℩crio 7314 (class class class)co 7358 Basecbs 17168 +_gcplusg 17209 lecple 17216 occoc 17217 Atomscatm 39720 HLchlt 39807 LHypclh 40441 LTrncltrn 40558 TEndoctendo 41209 DVecHcdvh 41535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-riotaBAD 39410

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-undef 8214 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-n0 12427 df-z 12514 df-uz 12778 df-fz 13451 df-struct 17106 df-slot 17141 df-ndx 17153 df-base 17169 df-plusg 17222 df-mulr 17223 df-sca 17225 df-vsca 17226 df-proset 18249 df-poset 18268 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-p1 18379 df-lat 18387 df-clat 18454 df-oposet 39633 df-ol 39635 df-oml 39636 df-covers 39723 df-ats 39724 df-atl 39755 df-cvlat 39779 df-hlat 39808 df-llines 39955 df-lplanes 39956 df-lvols 39957 df-lines 39958 df-psubsp 39960 df-pmap 39961 df-padd 40253 df-lhyp 40445 df-laut 40446 df-ldil 40561 df-ltrn 40562 df-trl 40616 df-tendo 41212 df-edring 41214 df-dvech 41536

This theorem is referenced by: dihordlem7 41671

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