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Mirrors > Home > MPE Home > Th. List > Mathboxes > dian0 | Structured version Visualization version GIF version |
Description: The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.) |
Ref | Expression |
---|---|
dian0.b | ⊢ 𝐵 = (Base‘𝐾) |
dian0.l | ⊢ ≤ = (le‘𝐾) |
dian0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dian0.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dian0 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dian0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dian0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2610 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | idltrn 34454 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
5 | 4 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
6 | eqid 2610 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2610 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
8 | 1, 6, 2, 7 | trlid0 34481 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
9 | 8 | adantr 480 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
10 | hlatl 33665 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
11 | 10 | adantr 480 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ AtLat) |
12 | simpl 472 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → 𝑋 ∈ 𝐵) | |
13 | dian0.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
14 | 1, 13, 6 | atl0le 33609 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (0.‘𝐾) ≤ 𝑋) |
15 | 11, 12, 14 | syl2an 493 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (0.‘𝐾) ≤ 𝑋) |
16 | 9, 15 | eqbrtrd 4605 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) ≤ 𝑋) |
17 | dian0.i | . . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
18 | 1, 13, 2, 3, 7, 17 | diaelval 35340 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (( I ↾ 𝐵) ∈ (𝐼‘𝑋) ↔ (( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊) ∧ (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) ≤ 𝑋))) |
19 | 5, 16, 18 | mpbir2and 959 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ (𝐼‘𝑋)) |
20 | ne0i 3880 | . 2 ⊢ (( I ↾ 𝐵) ∈ (𝐼‘𝑋) → (𝐼‘𝑋) ≠ ∅) | |
21 | 19, 20 | syl 17 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 class class class wbr 4583 I cid 4948 ↾ cres 5040 ‘cfv 5804 Basecbs 15695 lecple 15775 0.cp0 16860 AtLatcal 33569 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 DIsoAcdia 35335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-disoa 35336 |
This theorem is referenced by: dialss 35353 dibn0 35460 |
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