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Mirrors > Home > MPE Home > Th. List > meetval2lem | Structured version Visualization version GIF version |
Description: Lemma for meetval2 17635 and meeteu 17636. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 17636 into meetlem 17637? |
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 5072 | . . 3 ⊢ (𝑦 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑋)) | |
2 | breq2 5072 | . . 3 ⊢ (𝑦 = 𝑌 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑌)) | |
3 | 1, 2 | ralprg 4634 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ↔ (𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌))) |
4 | breq2 5072 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑋)) | |
5 | breq2 5072 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑧 ≤ 𝑦 ↔ 𝑧 ≤ 𝑌)) | |
6 | 4, 5 | ralprg 4634 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 ↔ (𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌))) |
7 | 6 | imbi1d 344 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) |
8 | 7 | ralbidv 3199 | . 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {cpr 4571 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 meetcmee 17557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 |
This theorem is referenced by: meetval2 17635 meeteu 17636 |
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