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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 2821 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | idltrn 37301 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
5 | 4 | adantr 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ ((LTrn‘𝐾)‘𝑊)) |
6 | eqid 2821 | . . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
7 | eqid 2821 | . . . . . 6 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
8 | 1, 6, 2, 7 | trlid0 37327 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
9 | 8 | adantr 483 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (((trL‘𝐾)‘𝑊)‘( I ↾ 𝐵)) = (0.‘𝐾)) |
10 | hlatl 36511 | . . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat) | |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ AtLat) |
12 | simpl 485 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → 𝑋 ∈ 𝐵) | |
13 | dian0.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
14 | 1, 13, 6 | atl0le 36455 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝐵) → (0.‘𝐾) ≤ 𝑋) |
15 | 11, 12, 14 | syl2an 597 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (0.‘𝐾) ≤ 𝑋) |
16 | 9, 15 | eqbrtrd 5088 | . . 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 38184 | . . 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 711 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → ( I ↾ 𝐵) ∈ (𝐼‘𝑋)) |
20 | 19 | ne0d 4301 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 class class class wbr 5066 I cid 5459 ↾ cres 5557 ‘cfv 6355 Basecbs 16483 lecple 16572 0.cp0 17647 AtLatcal 36415 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 trLctrl 37309 DIsoAcdia 38179 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-disoa 38180 |
This theorem is referenced by: dialss 38197 dibn0 38304 |
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