Theorem dihord2cN 41218

Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihjust.b	⊢ 𝐵 = (Base‘𝐾)
dihjust.l	⊢ ≤ = (le‘𝐾)
dihjust.j	⊢ ∨ = (join‘𝐾)
dihjust.m	⊢ ∧ = (meet‘𝐾)
dihjust.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjust.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjust.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
dihjust.J	⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
dihjust.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjust.s	⊢ ⊕ = (LSSum‘𝑈)
dihord2c.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihord2c.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihord2c.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dihord2cN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)))

Distinct variable groups:   ∨ ,𝑓   ∧ ,𝑓   ⊕ ,𝑓   𝑓,ℎ,𝐴   𝑓,𝐼   𝑓,𝐽   𝑅,𝑓   𝐵,𝑓,ℎ   𝑓,𝐻,ℎ   𝑓,𝐾,ℎ   ≤ ,𝑓,ℎ   𝑇,𝑓,ℎ   𝑓,𝑊,ℎ   𝑓,𝑋

Allowed substitution hints:   ⊕ (ℎ)   𝑅(ℎ)   𝑈(𝑓,ℎ)   𝐼(ℎ)   𝐽(ℎ)   ∨ (ℎ)   ∧ (ℎ)   𝑂(𝑓,ℎ)   𝑋(ℎ)

Proof of Theorem dihord2cN

Step	Hyp	Ref	Expression
1		simp3 1139	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)))
2		eqidd 2738	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑂 = 𝑂)
3		simp1 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		simp1l 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ HL)
5	4	hllatd 39360	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ Lat)
6		simp2 1138	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑋 ∈ 𝐵)
7		simp1r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐻)
8		dihjust.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
9		dihjust.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
10	8, 9	lhpbase 39995	. . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
11	7, 10	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐵)
12		dihjust.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
13	8, 12	latmcl 18507	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
14	5, 6, 11, 13	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑋 ∧ 𝑊) ∈ 𝐵)
15		dihjust.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
16	8, 15, 12	latmle2 18532	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
17	5, 6, 11, 16	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑋 ∧ 𝑊) ≤ 𝑊)
18		dihord2c.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
19		dihord2c.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
20		dihord2c.o	. . . 4 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
21		dihjust.i	. . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
22	8, 15, 9, 18, 19, 20, 21	dibopelval3 41145	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑂 = 𝑂)))
23	3, 14, 17, 22	syl12anc 837	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)) ↔ ((𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑂 = 𝑂)))
24	1, 2, 23	mpbir2and 713	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 ∧ 𝑊)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108  ⟨cop 4640   class class class wbr 5151   ↦ cmpt 5234   I cid 5586   ↾ cres 5695  ‘cfv 6569  (class class class)co 7438  Basecbs 17254  lecple 17314  joincjn 18378  meetcmee 18379  Latclat 18498  LSSumclsm 19676  Atomscatm 39259  HLchlt 39346  LHypclh 39981  LTrncltrn 40098  trLctrl 40155  DVecHcdvh 41075  DIsoBcdib 41135  DIsoCcdic 41169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  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 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  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-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-iun 5001  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-riota 7395  df-ov 7441  df-oprab 7442  df-lub 18413  df-glb 18414  df-join 18415  df-meet 18416  df-lat 18499  df-atl 39294  df-cvlat 39318  df-hlat 39347  df-lhyp 39985  df-disoa 41026  df-dib 41136

This theorem is referenced by: (None)