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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihwN | Structured version Visualization version GIF version |
Description: Value of isomorphism H at the fiducial hyperplane 𝑊. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihw.b | ⊢ 𝐵 = (Base‘𝐾) |
dihw.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihw.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihw.o | ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dihw.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihw.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
dihwN | ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihw.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | 1 | simprd 488 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
3 | dihw.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihw.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 36619 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
7 | 1 | simpld 487 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
8 | 7 | hllatd 35985 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
9 | eqid 2780 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9 | latref 17533 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵) → 𝑊(le‘𝐾)𝑊) |
11 | 8, 6, 10 | syl2anc 576 | . . . 4 ⊢ (𝜑 → 𝑊(le‘𝐾)𝑊) |
12 | 6, 11 | jca 504 | . . 3 ⊢ (𝜑 → (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) |
13 | dihw.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | eqid 2780 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 3, 9, 4, 13, 14 | dihvalb 37858 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
16 | 1, 12, 15 | syl2anc 576 | . 2 ⊢ (𝜑 → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
17 | dihw.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
18 | dihw.o | . . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
19 | eqid 2780 | . . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
20 | 3, 9, 4, 17, 18, 19, 14 | dibval2 37765 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
21 | 1, 12, 20 | syl2anc 576 | . 2 ⊢ (𝜑 → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
22 | eqid 2780 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
23 | 3, 9, 4, 17, 22, 19 | diaval 37653 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
24 | 1, 12, 23 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
25 | 9, 4, 17, 22 | trlle 36805 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
26 | 1, 25 | sylan 572 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
27 | 26 | ralrimiva 3134 | . . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
28 | rabid2 3322 | . . . . 5 ⊢ (𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊} ↔ ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) | |
29 | 27, 28 | sylibr 226 | . . . 4 ⊢ (𝜑 → 𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
30 | 24, 29 | eqtr4d 2819 | . . 3 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = 𝑇) |
31 | 30 | xpeq1d 5440 | . 2 ⊢ (𝜑 → ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 }) = (𝑇 × { 0 })) |
32 | 16, 21, 31 | 3eqtrd 2820 | 1 ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3090 {crab 3094 {csn 4444 class class class wbr 4934 ↦ cmpt 5013 I cid 5315 × cxp 5409 ↾ cres 5413 ‘cfv 6193 Basecbs 16345 lecple 16434 Latclat 17525 HLchlt 35971 LHypclh 36605 LTrncltrn 36722 trLctrl 36779 DIsoAcdia 37649 DIsoBcdib 37759 DIsoHcdih 37849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-riota 6943 df-ov 6985 df-oprab 6986 df-mpo 6987 df-map 8214 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-oposet 35797 df-ol 35799 df-oml 35800 df-covers 35887 df-ats 35888 df-atl 35919 df-cvlat 35943 df-hlat 35972 df-lhyp 36609 df-laut 36610 df-ldil 36725 df-ltrn 36726 df-trl 36780 df-disoa 37650 df-dib 37760 df-dih 37850 |
This theorem is referenced by: (None) |
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