Theorem dihord2cN 41560

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 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ HL)
5	4	hllatd 39703	. . . 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 1200	. . . . 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 40337	. . . . 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 18368	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
14	5, 6, 11, 13	syl3anc 1374	. . 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 18393	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
17	5, 6, 11, 16	syl3anc 1374	. . 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 41487	. . 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 714	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 1542   ∈ wcel 2114  ⟨cop 4587   class class class wbr 5099   ↦ cmpt 5180   I cid 5519   ↾ cres 5627  ‘cfv 6493  (class class class)co 7361  Basecbs 17141  lecple 17189  joincjn 18239  meetcmee 18240  Latclat 18359  LSSumclsm 19568  Atomscatm 39602  HLchlt 39689  LHypclh 40323  LTrncltrn 40440  trLctrl 40497  DVecHcdvh 41417  DIsoBcdib 41477  DIsoCcdic 41511

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-lub 18272  df-glb 18273  df-join 18274  df-meet 18275  df-lat 18360  df-atl 39637  df-cvlat 39661  df-hlat 39690  df-lhyp 40327  df-disoa 41368  df-dib 41478

This theorem is referenced by: (None)