Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihord2cN | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
dihord2cN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 〈𝑓, 𝑂〉 ∈ (𝐼‘(𝑋 ∧ 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 1134 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) | |
2 | eqidd 2824 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑂 = 𝑂) | |
3 | simp1 1132 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | simp1l 1193 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ HL) | |
5 | 4 | hllatd 36502 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ Lat) |
6 | simp2 1133 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑋 ∈ 𝐵) | |
7 | simp1r 1194 | . . . . 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 37136 | . . . . 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 17664 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵) |
14 | 5, 6, 11, 13 | syl3anc 1367 | . . 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 17689 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊) |
17 | 5, 6, 11, 16 | syl3anc 1367 | . . 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 38286 | . . 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 834 | . 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 711 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 〈𝑓, 𝑂〉 ∈ (𝐼‘(𝑋 ∧ 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 ↦ cmpt 5148 I cid 5461 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 joincjn 17556 meetcmee 17557 Latclat 17657 LSSumclsm 18761 Atomscatm 36401 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 DVecHcdvh 38216 DIsoBcdib 38276 DIsoCcdic 38310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-lat 17658 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 df-disoa 38167 df-dib 38277 |
This theorem is referenced by: (None) |
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