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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 495 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
3 | dihw.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihw.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 39359 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
7 | 1 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
8 | 7 | hllatd 38724 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
9 | eqid 2724 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9 | latref 18396 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵) → 𝑊(le‘𝐾)𝑊) |
11 | 8, 6, 10 | syl2anc 583 | . . . 4 ⊢ (𝜑 → 𝑊(le‘𝐾)𝑊) |
12 | 6, 11 | jca 511 | . . 3 ⊢ (𝜑 → (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) |
13 | dihw.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | eqid 2724 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 3, 9, 4, 13, 14 | dihvalb 40598 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
16 | 1, 12, 15 | syl2anc 583 | . 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 2724 | . . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
20 | 3, 9, 4, 17, 18, 19, 14 | dibval2 40505 | . . 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 583 | . 2 ⊢ (𝜑 → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
22 | eqid 2724 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
23 | 3, 9, 4, 17, 22, 19 | diaval 40393 | . . . . 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 583 | . . . 4 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
25 | 9, 4, 17, 22 | trlle 39545 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
26 | 1, 25 | sylan 579 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
27 | 26 | ralrimiva 3138 | . . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
28 | rabid2 3456 | . . . . 5 ⊢ (𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊} ↔ ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) | |
29 | 27, 28 | sylibr 233 | . . . 4 ⊢ (𝜑 → 𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
30 | 24, 29 | eqtr4d 2767 | . . 3 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = 𝑇) |
31 | 30 | xpeq1d 5695 | . 2 ⊢ (𝜑 → ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 }) = (𝑇 × { 0 })) |
32 | 16, 21, 31 | 3eqtrd 2768 | 1 ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 {crab 3424 {csn 4620 class class class wbr 5138 ↦ cmpt 5221 I cid 5563 × cxp 5664 ↾ cres 5668 ‘cfv 6533 Basecbs 17143 lecple 17203 Latclat 18386 HLchlt 38710 LHypclh 39345 LTrncltrn 39462 trLctrl 39519 DIsoAcdia 40389 DIsoBcdib 40499 DIsoHcdih 40589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8818 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-oposet 38536 df-ol 38538 df-oml 38539 df-covers 38626 df-ats 38627 df-atl 38658 df-cvlat 38682 df-hlat 38711 df-lhyp 39349 df-laut 39350 df-ldil 39465 df-ltrn 39466 df-trl 39520 df-disoa 40390 df-dib 40500 df-dih 40590 |
This theorem is referenced by: (None) |
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