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Mirrors > Home > MPE Home > Th. List > joinval2lem | Structured version Visualization version GIF version |
Description: Lemma for joinval2 17607 and joineu 17608. (Contributed by NM, 12-Sep-2018.) TODO: combine this through joineu 17608 into joinlem 17609? |
Ref | Expression |
---|---|
joinval2.b | ⊢ 𝐵 = (Base‘𝐾) |
joinval2.l | ⊢ ≤ = (le‘𝐾) |
joinval2.j | ⊢ ∨ = (join‘𝐾) |
joinval2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joinval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
joinval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
joinval2lem | ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5060 | . . 3 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑥 ↔ 𝑋 ≤ 𝑥)) | |
2 | breq1 5060 | . . 3 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑥 ↔ 𝑌 ≤ 𝑥)) | |
3 | 1, 2 | ralprg 4624 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ↔ (𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥))) |
4 | breq1 5060 | . . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ 𝑧 ↔ 𝑋 ≤ 𝑧)) | |
5 | breq1 5060 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑦 ≤ 𝑧 ↔ 𝑌 ≤ 𝑧)) | |
6 | 4, 5 | ralprg 4624 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 ↔ (𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧))) |
7 | 6 | imbi1d 343 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) |
8 | 7 | ralbidv 3194 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) |
9 | 3, 8 | anbi12d 630 | 1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {cpr 4559 class class class wbr 5057 ‘cfv 6348 Basecbs 16471 lecple 16560 joincjn 17542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 |
This theorem is referenced by: joinval2 17607 joineu 17608 |
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