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Mirrors > Home > MPE Home > Th. List > eqsupd | Structured version Visualization version GIF version |
Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
Ref | Expression |
---|---|
supmo.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
eqsupd.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
eqsupd.3 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) |
eqsupd.4 | ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) |
Ref | Expression |
---|---|
eqsupd | ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsupd.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
2 | eqsupd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐶𝑅𝑦) | |
3 | 2 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦) |
4 | eqsupd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) | |
5 | 4 | expr 641 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
6 | 5 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
7 | supmo.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
8 | 7 | eqsup 8245 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) |
9 | 1, 3, 6, 8 | mp3and 1419 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 Or wor 4958 supcsup 8229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-po 4959 df-so 4960 df-iota 5768 df-riota 6511 df-sup 8231 |
This theorem is referenced by: supmax 8256 supiso 8264 dfgcd2 15101 esumpcvgval 29467 esum2d 29482 mblfinlem3 32618 mblfinlem4 32619 ismblfin 32620 itg2addnclem 32631 radcnvrat 37535 |
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