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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 5674 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | 1, 2 | sylib 208 | . . 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 8349 | . . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
9 | 7, 8 | syl6eq 2672 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ◡𝑅)) |
10 | 3, 4, 5, 6, 9 | supgtoreq 8376 | . 2 ⊢ (𝜑 → (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆)) |
11 | ne0i 3921 | . . . . . . 7 ⊢ (𝐶 ∈ 𝐵 → 𝐵 ≠ ∅) | |
12 | 6, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
13 | fiinfcl 8407 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
14 | 1, 5, 12, 4, 13 | syl13anc 1328 | . . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
15 | 7, 14 | eqeltrd 2701 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
16 | brcnvg 5303 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶◡𝑅𝑆 ↔ 𝑆𝑅𝐶)) | |
17 | 16 | bicomd 213 | . . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
18 | 6, 15, 17 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
19 | 18 | orbi1d 739 | . 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆))) |
20 | 10, 19 | mpbird 247 | 1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 Or wor 5034 ◡ccnv 5113 Fincfn 7955 supcsup 8346 infcinf 8347 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-om 7066 df-1o 7560 df-er 7742 df-en 7956 df-fin 7959 df-sup 8348 df-inf 8349 |
This theorem is referenced by: (None) |
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