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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihopelvalbN | Structured version Visualization version GIF version |
Description: Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihval3.b | ⊢ 𝐵 = (Base‘𝐾) |
dihval3.l | ⊢ ≤ = (le‘𝐾) |
dihval3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihval3.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihval3.r | ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) |
dihval3.o | ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dihval3.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihopelvalbN | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihval3.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihval3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | dihval3.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dihval3.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
5 | eqid 2798 | . . . 4 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dihvalb 38533 | . . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋)) |
7 | 6 | eleq2d 2875 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ 〈𝐹, 𝑆〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋))) |
8 | dihval3.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
9 | dihval3.r | . . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) | |
10 | dihval3.o | . . 3 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | 1, 2, 3, 8, 9, 10, 5 | dibopelval3 38444 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
12 | 7, 11 | bitrd 282 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 𝑂))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 ↦ cmpt 5110 I cid 5424 ↾ cres 5521 ‘cfv 6324 Basecbs 16475 lecple 16564 LHypclh 37280 LTrncltrn 37397 trLctrl 37454 DIsoBcdib 38434 DIsoHcdih 38524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-disoa 38325 df-dib 38435 df-dih 38525 |
This theorem is referenced by: dihmeetlem1N 38586 dihglblem5apreN 38587 dihmeetlem4preN 38602 |
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