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Mirrors > Home > MPE Home > Th. List > infltoreq | Structured version Visualization version GIF version |
Description: The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
infltoreq.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infltoreq.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
infltoreq.3 | ⊢ (𝜑 → 𝐵 ∈ Fin) |
infltoreq.4 | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
infltoreq.5 | ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) |
Ref | Expression |
---|---|
infltoreq | ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infltoreq.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
2 | cnvso 6287 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
4 | infltoreq.2 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
5 | infltoreq.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) | |
6 | infltoreq.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵) | |
7 | infltoreq.5 | . . . 4 ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅)) | |
8 | df-inf 9466 | . . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
9 | 7, 8 | eqtrdi 2781 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ◡𝑅)) |
10 | 3, 4, 5, 6, 9 | supgtoreq 9493 | . 2 ⊢ (𝜑 → (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆)) |
11 | 6 | ne0d 4331 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
12 | fiinfcl 9524 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
13 | 1, 5, 11, 4, 12 | syl13anc 1369 | . . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
14 | 7, 13 | eqeltrd 2825 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
15 | brcnvg 5876 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶◡𝑅𝑆 ↔ 𝑆𝑅𝐶)) | |
16 | 15 | bicomd 222 | . . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
17 | 6, 14, 16 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
18 | 17 | orbi1d 914 | . 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆))) |
19 | 10, 18 | mpbird 256 | 1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2930 ⊆ wss 3939 ∅c0 4318 class class class wbr 5143 Or wor 5583 ◡ccnv 5671 Fincfn 8962 supcsup 9463 infcinf 9464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-om 7869 df-en 8963 df-fin 8966 df-sup 9465 df-inf 9466 |
This theorem is referenced by: (None) |
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