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| Mirrors > Home > MPE Home > Th. List > Mathboxes > diameetN | Structured version Visualization version GIF version | ||
| Description: Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| diam.m | ⊢ ∧ = (meet‘𝐾) | 
| diam.h | ⊢ 𝐻 = (LHyp‘𝐾) | 
| diam.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | 
| Ref | Expression | 
|---|---|
| diameetN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 2 | diam.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
| 3 | simpll 766 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝐾 ∈ HL) | |
| 4 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 5 | diam.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
| 6 | diam.i | . . . . . 6 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
| 7 | 4, 5, 6 | diadmclN 41040 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ (Base‘𝐾)) | 
| 8 | 7 | adantrr 717 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑋 ∈ (Base‘𝐾)) | 
| 9 | 4, 5, 6 | diadmclN 41040 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ dom 𝐼) → 𝑌 ∈ (Base‘𝐾)) | 
| 10 | 9 | adantrl 716 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → 𝑌 ∈ (Base‘𝐾)) | 
| 11 | 1, 2, 3, 8, 10 | meetval 18437 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) | 
| 12 | 11 | fveq2d 6909 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∧ 𝑌)) = (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌}))) | 
| 13 | simpl 482 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 14 | prssi 4820 | . . . 4 ⊢ ((𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼) → {𝑋, 𝑌} ⊆ dom 𝐼) | |
| 15 | 14 | adantl 481 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → {𝑋, 𝑌} ⊆ dom 𝐼) | 
| 16 | prnzg 4777 | . . . 4 ⊢ (𝑋 ∈ dom 𝐼 → {𝑋, 𝑌} ≠ ∅) | |
| 17 | 16 | ad2antrl 728 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → {𝑋, 𝑌} ≠ ∅) | 
| 18 | 1, 5, 6 | diaglbN 41058 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑋, 𝑌} ⊆ dom 𝐼 ∧ {𝑋, 𝑌} ≠ ∅)) → (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥)) | 
| 19 | 13, 15, 17, 18 | syl12anc 836 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥)) | 
| 20 | fveq2 6905 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘𝑥) = (𝐼‘𝑋)) | |
| 21 | fveq2 6905 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝐼‘𝑥) = (𝐼‘𝑌)) | |
| 22 | 20, 21 | iinxprg 5088 | . . 3 ⊢ ((𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | 
| 23 | 22 | adantl 481 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | 
| 24 | 12, 19, 23 | 3eqtrd 2780 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 {cpr 4627 ∩ ciin 4991 dom cdm 5684 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 glbcglb 18357 meetcmee 18359 HLchlt 39352 LHypclh 39987 DIsoAcdia 41031 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-p1 18472 df-lat 18478 df-clat 18545 df-oposet 39178 df-ol 39180 df-oml 39181 df-covers 39268 df-ats 39269 df-atl 39300 df-cvlat 39324 df-hlat 39353 df-lhyp 39991 df-laut 39992 df-ldil 40107 df-ltrn 40108 df-trl 40162 df-disoa 41032 | 
| This theorem is referenced by: diainN 41060 djajN 41140 | 
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