Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pltle | Structured version Visualization version GIF version |
Description: Less-than implies less-than-or-equal. (pssss 3664 analog.) (Contributed by NM, 4-Dec-2011.) |
Ref | Expression |
---|---|
pltval.l | ⊢ ≤ = (le‘𝐾) |
pltval.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltle | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | pltval.s | . . . 4 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltval 16783 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
4 | 3 | simprbda 651 | . 2 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 < 𝑌) → 𝑋 ≤ 𝑌) |
5 | 4 | ex 449 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 → 𝑋 ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 ‘cfv 5804 lecple 15775 ltcplt 16764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-plt 16781 |
This theorem is referenced by: pleval2 16788 pltnlt 16791 pltn2lp 16792 plttr 16793 pospo 16796 ogrpaddlt 29049 isarchi3 29072 archirngz 29074 archiabllem2a 29079 orngsqr 29135 ornglmullt 29138 orngrmullt 29139 atnlt 33618 cvlcvr1 33644 hlrelat 33706 hlrelat3 33716 cvratlem 33725 atltcvr 33739 atlelt 33742 llnnlt 33827 lplnnle2at 33845 lplnnlt 33869 lvolnle3at 33886 lvolnltN 33922 cdlemblem 34097 cdlemb 34098 lhpexle1 34312 |
Copyright terms: Public domain | W3C validator |