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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 2966 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦) |
4 | eqsupd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝐶)) → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) | |
5 | 4 | expr 643 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
6 | 5 | ralrimiva 2966 | . 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) |
7 | supmo.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
8 | 7 | eqsup 8362 | . 2 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → sup(𝐵, 𝐴, 𝑅) = 𝐶)) |
9 | 1, 3, 6, 8 | mp3and 1427 | 1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 Or wor 5034 supcsup 8346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-po 5035 df-so 5036 df-iota 5851 df-riota 6611 df-sup 8348 |
This theorem is referenced by: supmax 8373 supiso 8381 dfgcd2 15263 esumpcvgval 30140 esum2d 30155 mblfinlem3 33448 mblfinlem4 33449 ismblfin 33450 itg2addnclem 33461 radcnvrat 38513 |
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