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Mirrors > Home > MPE Home > Th. List > latmlem12 | Structured version Visualization version GIF version |
Description: Add join to both sides of a lattice ordering. (ss2in 3840 analog.) (Contributed by NM, 10-Nov-2011.) |
Ref | Expression |
---|---|
latmle.b | ⊢ 𝐵 = (Base‘𝐾) |
latmle.l | ⊢ ≤ = (le‘𝐾) |
latmle.m | ⊢ ∧ = (meet‘𝐾) |
Ref | Expression |
---|---|
latmlem12 | ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑍 ≤ 𝑊) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐾 ∈ Lat) | |
2 | simp2l 1087 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
3 | simp2r 1088 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
4 | simp3l 1089 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑍 ∈ 𝐵) | |
5 | latmle.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
6 | latmle.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
7 | latmle.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
8 | 5, 6, 7 | latmlem1 17081 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍))) |
9 | 1, 2, 3, 4, 8 | syl13anc 1328 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍))) |
10 | simp3r 1090 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵) | |
11 | 5, 6, 7 | latmlem2 17082 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑍 ≤ 𝑊 → (𝑌 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
12 | 1, 4, 10, 3, 11 | syl13anc 1328 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑍 ≤ 𝑊 → (𝑌 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
13 | 5, 7 | latmcl 17052 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 ∧ 𝑍) ∈ 𝐵) |
14 | 1, 2, 4, 13 | syl3anc 1326 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑋 ∧ 𝑍) ∈ 𝐵) |
15 | 5, 7 | latmcl 17052 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ∧ 𝑍) ∈ 𝐵) |
16 | 1, 3, 4, 15 | syl3anc 1326 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑌 ∧ 𝑍) ∈ 𝐵) |
17 | 5, 7 | latmcl 17052 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵) |
18 | 1, 3, 10, 17 | syl3anc 1326 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑌 ∧ 𝑊) ∈ 𝐵) |
19 | 5, 6 | lattr 17056 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ ((𝑋 ∧ 𝑍) ∈ 𝐵 ∧ (𝑌 ∧ 𝑍) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ∈ 𝐵)) → (((𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍) ∧ (𝑌 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊)) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
20 | 1, 14, 16, 18, 19 | syl13anc 1328 | . 2 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍) ∧ (𝑌 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊)) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
21 | 9, 12, 20 | syl2and 500 | 1 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑍 ≤ 𝑊) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 lecple 15948 meetcmee 16945 Latclat 17045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 |
This theorem is referenced by: dalem10 34959 dalem55 35013 dalawlem3 35159 dalawlem7 35163 dalawlem11 35167 dalawlem12 35168 cdlemk51 36241 |
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