Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 498 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
3 | dihw.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihw.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 37136 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
6 | 2, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵) |
7 | 1 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
8 | 7 | hllatd 36502 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
9 | eqid 2823 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9 | latref 17665 | . . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑊 ∈ 𝐵) → 𝑊(le‘𝐾)𝑊) |
11 | 8, 6, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑊(le‘𝐾)𝑊) |
12 | 6, 11 | jca 514 | . . 3 ⊢ (𝜑 → (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) |
13 | dihw.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
14 | eqid 2823 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 3, 9, 4, 13, 14 | dihvalb 38375 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑊 ∈ 𝐵 ∧ 𝑊(le‘𝐾)𝑊)) → (𝐼‘𝑊) = (((DIsoB‘𝐾)‘𝑊)‘𝑊)) |
16 | 1, 12, 15 | syl2anc 586 | . 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 2823 | . . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
20 | 3, 9, 4, 17, 18, 19, 14 | dibval2 38282 | . . 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 586 | . 2 ⊢ (𝜑 → (((DIsoB‘𝐾)‘𝑊)‘𝑊) = ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 })) |
22 | eqid 2823 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
23 | 3, 9, 4, 17, 22, 19 | diaval 38170 | . . . . 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 586 | . . . 4 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
25 | 9, 4, 17, 22 | trlle 37322 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
26 | 1, 25 | sylan 582 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
27 | 26 | ralrimiva 3184 | . . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
28 | rabid2 3383 | . . . . 5 ⊢ (𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊} ↔ ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) | |
29 | 27, 28 | sylibr 236 | . . . 4 ⊢ (𝜑 → 𝑇 = {𝑔 ∈ 𝑇 ∣ (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊}) |
30 | 24, 29 | eqtr4d 2861 | . . 3 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = 𝑇) |
31 | 30 | xpeq1d 5586 | . 2 ⊢ (𝜑 → ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 }) = (𝑇 × { 0 })) |
32 | 16, 21, 31 | 3eqtrd 2862 | 1 ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 {csn 4569 class class class wbr 5068 ↦ cmpt 5148 I cid 5461 × cxp 5555 ↾ cres 5559 ‘cfv 6357 Basecbs 16485 lecple 16574 Latclat 17657 HLchlt 36488 LHypclh 37122 LTrncltrn 37239 trLctrl 37296 DIsoAcdia 38166 DIsoBcdib 38276 DIsoHcdih 38366 |
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-mpo 7163 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-disoa 38167 df-dib 38277 df-dih 38367 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |