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 479 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
3 | dihw.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihw.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 35284 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
7 | 1 | simpld 475 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
8 | hllat 34650 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
10 | eqid 2622 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
11 | 3, 10 | latref 17053 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵) → 𝑊(le‘𝐾)𝑊) |
12 | 9, 6, 11 | syl2anc 693 | . . . 4 ⊢ (𝜑 → 𝑊(le‘𝐾)𝑊) |
13 | 6, 12 | jca 554 | . . 3 ⊢ (𝜑 → (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) |
14 | dihw.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
15 | eqid 2622 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
16 | 3, 10, 4, 14, 15 | dihvalb 36526 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
17 | 1, 13, 16 | syl2anc 693 | . 2 ⊢ (𝜑 → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
18 | dihw.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
19 | dihw.o | . . . 4 ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
20 | eqid 2622 | . . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
21 | 3, 10, 4, 18, 19, 20, 15 | dibval2 36433 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
22 | 1, 13, 21 | syl2anc 693 | . 2 ⊢ (𝜑 → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
23 | eqid 2622 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
24 | 3, 10, 4, 18, 23, 20 | diaval 36321 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
25 | 1, 13, 24 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
26 | 10, 4, 18, 23 | trlle 35471 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
27 | 1, 26 | sylan 488 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
28 | 27 | ralrimiva 2966 | . . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
29 | rabid2 3118 | . . . . 5 ⊢ (𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊} ↔ ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) | |
30 | 28, 29 | sylibr 224 | . . . 4 ⊢ (𝜑 → 𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
31 | 25, 30 | eqtr4d 2659 | . . 3 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = 𝑇) |
32 | 31 | xpeq1d 5138 | . 2 ⊢ (𝜑 → ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 }) = (𝑇 × { 0 })) |
33 | 17, 22, 32 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 {csn 4177 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ↾ cres 5116 ‘cfv 5888 Basecbs 15857 lecple 15948 Latclat 17045 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 trLctrl 35445 DIsoAcdia 36317 DIsoBcdib 36427 DIsoHcdih 36517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-disoa 36318 df-dib 36428 df-dih 36518 |
This theorem is referenced by: (None) |
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