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Mirrors > Home > MPE Home > Th. List > lemeet1 | Structured version Visualization version GIF version |
Description: A meet's first argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011.) (Revised by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
meetval2.b | ⊢ 𝐵 = (Base‘𝐾) |
meetval2.l | ⊢ ≤ = (le‘𝐾) |
meetval2.m | ⊢ ∧ = (meet‘𝐾) |
meetval2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
meetval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
meetval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
meetlem.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
Ref | Expression |
---|---|
lemeet1 | ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetval2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | meetval2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | meetval2.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
4 | meetval2.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | meetval2.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | meetval2.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | meetlem.e | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) | |
8 | 1, 2, 3, 4, 5, 6, 7 | meetlem 17634 | . 2 ⊢ (𝜑 → (((𝑋 ∧ 𝑌) ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ (𝑋 ∧ 𝑌)))) |
9 | 8 | simplld 766 | 1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 〈cop 4572 class class class wbr 5065 dom cdm 5554 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 lecple 16571 meetcmee 17554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-glb 17584 df-meet 17586 |
This theorem is referenced by: meetle 17637 latmle1 17685 |
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