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 498 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
3 | dihw.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
4 | dihw.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | 3, 4 | lhpbase 37167 | . . . . 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 36533 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat) |
9 | eqid 2820 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9 | latref 17658 | . . . . 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 2820 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
15 | 3, 9, 4, 13, 14 | dihvalb 38406 | . . 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 2820 | . . . 4 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
20 | 3, 9, 4, 17, 18, 19, 14 | dibval2 38313 | . . 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 2820 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
23 | 3, 9, 4, 17, 22, 19 | diaval 38201 | . . . . 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 37353 | . . . . . . 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 3181 | . . . . 5 ⊢ (𝜑 → ∀𝑔 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑔)(le‘𝐾)𝑊) |
28 | rabid2 3380 | . . . . 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 2858 | . . 3 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘𝑊) = 𝑇) |
31 | 30 | xpeq1d 5577 | . 2 ⊢ (𝜑 → ((((DIsoA‘𝐾)‘𝑊)‘𝑊) × { 0 }) = (𝑇 × { 0 })) |
32 | 16, 21, 31 | 3eqtrd 2859 | 1 ⊢ (𝜑 → (𝐼‘𝑊) = (𝑇 × { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 {crab 3141 {csn 4560 class class class wbr 5059 ↦ cmpt 5139 I cid 5452 × cxp 5546 ↾ cres 5550 ‘cfv 6348 Basecbs 16478 lecple 16567 Latclat 17650 HLchlt 36519 LHypclh 37153 LTrncltrn 37270 trLctrl 37327 DIsoAcdia 38197 DIsoBcdib 38307 DIsoHcdih 38397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-map 8401 df-proset 17533 df-poset 17551 df-plt 17563 df-lub 17579 df-glb 17580 df-join 17581 df-meet 17582 df-p0 17644 df-p1 17645 df-lat 17651 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-atl 36467 df-cvlat 36491 df-hlat 36520 df-lhyp 37157 df-laut 37158 df-ldil 37273 df-ltrn 37274 df-trl 37328 df-disoa 38198 df-dib 38308 df-dih 38398 |
This theorem is referenced by: (None) |
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