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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 6133 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
3 | 1, 2 | sylib 220 | . . 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 8901 | . . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
9 | 7, 8 | syl6eq 2872 | . . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ◡𝑅)) |
10 | 3, 4, 5, 6, 9 | supgtoreq 8928 | . 2 ⊢ (𝜑 → (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆)) |
11 | 6 | ne0d 4300 | . . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅) |
12 | fiinfcl 8959 | . . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) | |
13 | 1, 5, 11, 4, 12 | syl13anc 1368 | . . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵) |
14 | 7, 13 | eqeltrd 2913 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵) |
15 | brcnvg 5744 | . . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶◡𝑅𝑆 ↔ 𝑆𝑅𝐶)) | |
16 | 15 | bicomd 225 | . . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
17 | 6, 14, 16 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶◡𝑅𝑆)) |
18 | 17 | orbi1d 913 | . 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶◡𝑅𝑆 ∨ 𝐶 = 𝑆))) |
19 | 10, 18 | mpbird 259 | 1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 Or wor 5467 ◡ccnv 5548 Fincfn 8503 supcsup 8898 infcinf 8899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-om 7575 df-1o 8096 df-er 8283 df-en 8504 df-fin 8507 df-sup 8900 df-inf 8901 |
This theorem is referenced by: (None) |
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