Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pltval | Structured version Visualization version GIF version |
Description: Less-than relation. (df-pss 3556 analog.) (Contributed by NM, 12-Oct-2011.) |
Ref | Expression |
---|---|
pltval.l | ⊢ ≤ = (le‘𝐾) |
pltval.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltval | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
2 | pltval.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltfval 16782 | . . . 4 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
4 | 3 | breqd 4594 | . . 3 ⊢ (𝐾 ∈ 𝐴 → (𝑋 < 𝑌 ↔ 𝑋( ≤ ∖ I )𝑌)) |
5 | brdif 4635 | . . . 4 ⊢ (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌)) | |
6 | ideqg 5195 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐶 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
7 | 6 | necon3bbid 2819 | . . . . . 6 ⊢ (𝑌 ∈ 𝐶 → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (¬ 𝑋 I 𝑌 ↔ 𝑋 ≠ 𝑌)) |
9 | 8 | anbi2d 736 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑋 I 𝑌) ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
10 | 5, 9 | syl5bb 271 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋( ≤ ∖ I )𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
11 | 4, 10 | sylan9bb 732 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
12 | 11 | 3impb 1252 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐶) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 class class class wbr 4583 I cid 4948 ‘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: pltle 16784 pltne 16785 pleval2i 16787 pltnle 16789 pltval3 16790 plttr 16793 latnlemlt 16907 latnle 16908 ipolt 16982 ogrpaddlt 29049 ogrpsublt 29053 ornglmullt 29138 orngrmullt 29139 orngmullt 29140 ofldlt1 29144 opltn0 33495 cvrval2 33579 cvrnbtwn2 33580 cvrnbtwn3 33581 cvrle 33583 cvrnbtwn4 33584 cvrne 33586 atlltn0 33611 hlrelat5N 33705 llnle 33822 lplnle 33844 llncvrlpln2 33861 lplncvrlvol2 33919 lhp2lt 34305 lautlt 34395 |
Copyright terms: Public domain | W3C validator |