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Mirrors > Home > MPE Home > Th. List > p0le | Structured version Visualization version GIF version |
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.) |
Ref | Expression |
---|---|
p0le.b | ⊢ 𝐵 = (Base‘𝐾) |
p0le.g | ⊢ 𝐺 = (glb‘𝐾) |
p0le.l | ⊢ ≤ = (le‘𝐾) |
p0le.0 | ⊢ 0 = (0.‘𝐾) |
p0le.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
p0le.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
p0le.d | ⊢ (𝜑 → 𝐵 ∈ dom 𝐺) |
Ref | Expression |
---|---|
p0le | ⊢ (𝜑 → 0 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | p0le.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
2 | p0le.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
3 | p0le.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
4 | p0le.0 | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 2, 3, 4 | p0val 17041 | . . 3 ⊢ (𝐾 ∈ 𝑉 → 0 = (𝐺‘𝐵)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → 0 = (𝐺‘𝐵)) |
7 | p0le.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
8 | p0le.d | . . 3 ⊢ (𝜑 → 𝐵 ∈ dom 𝐺) | |
9 | p0le.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | 2, 7, 3, 1, 8, 9 | glble 17000 | . 2 ⊢ (𝜑 → (𝐺‘𝐵) ≤ 𝑋) |
11 | 6, 10 | eqbrtrd 4675 | 1 ⊢ (𝜑 → 0 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 Basecbs 15857 lecple 15948 glbcglb 16943 0.cp0 17037 |
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-glb 16975 df-p0 17039 |
This theorem is referenced by: op0le 34473 atl0le 34591 |
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