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Mirrors > Home > MPE Home > Th. List > meetval2lem | Structured version Visualization version GIF version |
Description: Lemma for meetval2 18453 and meeteu 18454. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18454 into meetlem 18455? |
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) |
Ref | Expression |
---|---|
meetval2lem | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5152 | . . 3 ⊢ (𝑦 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑋)) | |
2 | breq2 5152 | . . 3 ⊢ (𝑦 = 𝑌 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑌)) | |
3 | 1, 2 | ralprg 4701 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ↔ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌))) |
4 | breq2 5152 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑋)) | |
5 | breq2 5152 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑌)) | |
6 | 4, 5 | ralprg 4701 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 ↔ (𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌))) |
7 | 6 | imbi1d 341 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) |
8 | 7 | ralbidv 3176 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) |
9 | 3, 8 | anbi12d 632 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {cpr 4633 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 lecple 17305 meetcmee 18370 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 |
This theorem is referenced by: meetval2 18453 meeteu 18454 meetdm3 48768 |
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