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Mirrors > Home > MPE Home > Th. List > Mathboxes > op1le | Structured version Visualization version GIF version |
Description: If the orthoposet unit is less than or equal to an element, the element equals the unit. (chle0 28302 analog.) (Contributed by NM, 5-Dec-2011.) |
Ref | Expression |
---|---|
ople1.b | ⊢ 𝐵 = (Base‘𝐾) |
ople1.l | ⊢ ≤ = (le‘𝐾) |
ople1.u | ⊢ 1 = (1.‘𝐾) |
Ref | Expression |
---|---|
op1le | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ople1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ople1.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
3 | ople1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | ople1 34478 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 1 ) |
5 | 4 | biantrurd 529 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ (𝑋 ≤ 1 ∧ 1 ≤ 𝑋))) |
6 | opposet 34468 | . . . 4 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) |
8 | simpr 477 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 1, 3 | op1cl 34472 | . . . 4 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 1 ∈ 𝐵) |
11 | 1, 2 | posasymb 16952 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 1 ∈ 𝐵) → ((𝑋 ≤ 1 ∧ 1 ≤ 𝑋) ↔ 𝑋 = 1 )) |
12 | 7, 8, 10, 11 | syl3anc 1326 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ((𝑋 ≤ 1 ∧ 1 ≤ 𝑋) ↔ 𝑋 = 1 )) |
13 | 5, 12 | bitrd 268 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 ≤ 𝑋 ↔ 𝑋 = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 Posetcpo 16940 1.cp1 17038 OPcops 34459 |
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 |
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-preset 16928 df-poset 16946 df-lub 16974 df-p1 17040 df-oposet 34463 |
This theorem is referenced by: glb0N 34480 lhpj1 35308 |
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