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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrle | Structured version Visualization version GIF version |
Description: The covers relation implies the "less than or equal to" relation. (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
cvrle.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrle.l | ⊢ ≤ = (le‘𝐾) |
cvrle.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
cvrle | ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrle.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2823 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
3 | cvrle.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrlt 36408 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌) |
5 | cvrle.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | 5, 2 | pltval 17572 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋(lt‘𝐾)𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
7 | 6 | simprbda 501 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋(lt‘𝐾)𝑌) → 𝑋 ≤ 𝑌) |
8 | 4, 7 | syldan 593 | 1 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 ltcplt 17553 ⋖ ccvr 36400 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-plt 17570 df-covers 36404 |
This theorem is referenced by: cvrnbtwn4 36417 cvrcmp 36421 atcvrj2b 36570 atexchcvrN 36578 llncmp 36660 llncvrlpln 36696 lplncmp 36700 lplncvrlvol 36754 lvolcmp 36755 |
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