![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 2772 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | idltrn 36679 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
5 | 4 | adantr 473 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
6 | eqid 2772 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2772 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
8 | 1, 6, 2, 7 | trlid0 36705 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
9 | 8 | adantr 473 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
10 | hlatl 35889 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
11 | 10 | adantr 473 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ AtLat) |
12 | simpl 475 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → 𝑋 ∈ 𝐵) | |
13 | dian0.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
14 | 1, 13, 6 | atl0le 35833 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (0.‘𝐾) ≤ 𝑋) |
15 | 11, 12, 14 | syl2an 586 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (0.‘𝐾) ≤ 𝑋) |
16 | 9, 15 | eqbrtrd 4945 | . . 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 37562 | . . 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 700 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ (𝐼‘𝑋)) |
20 | 19 | ne0d 4182 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∅c0 4173 class class class wbr 4923 I cid 5304 ↾ cres 5402 ‘cfv 6182 Basecbs 16329 lecple 16418 0.cp0 17495 AtLatcal 35793 HLchlt 35879 LHypclh 36513 LTrncltrn 36630 trLctrl 36687 DIsoAcdia 37557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-map 8200 df-proset 17386 df-poset 17404 df-plt 17416 df-lub 17432 df-glb 17433 df-join 17434 df-meet 17435 df-p0 17497 df-p1 17498 df-lat 17504 df-clat 17566 df-oposet 35705 df-ol 35707 df-oml 35708 df-covers 35795 df-ats 35796 df-atl 35827 df-cvlat 35851 df-hlat 35880 df-lhyp 36517 df-laut 36518 df-ldil 36633 df-ltrn 36634 df-trl 36688 df-disoa 37558 |
This theorem is referenced by: dialss 37575 dibn0 37682 |
Copyright terms: Public domain | W3C validator |