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| Mirrors > Home > MPE Home > Th. List > infglb | Structured version Visualization version GIF version | ||
| Description: An infimum is the greatest lower bound. See also infcl 9528 and inflb 9529. (Contributed by AV, 3-Sep-2020.) |
| Ref | Expression |
|---|---|
| infcl.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
| infcl.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
| Ref | Expression |
|---|---|
| infglb | ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-inf 9483 | . . . . 5 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
| 2 | 1 | breq1i 5150 | . . . 4 ⊢ (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶) |
| 3 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴) | |
| 4 | infcl.1 | . . . . . . . 8 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 5 | cnvso 6308 | . . . . . . . 8 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
| 6 | 4, 5 | sylib 218 | . . . . . . 7 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
| 7 | infcl.2 | . . . . . . . 8 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
| 8 | 4, 7 | infcllem 9527 | . . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
| 9 | 6, 8 | supcl 9498 | . . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
| 10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
| 11 | brcnvg 5890 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶)) | |
| 12 | 11 | bicomd 223 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅))) |
| 13 | 3, 10, 12 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅))) |
| 14 | 2, 13 | bitrid 283 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅))) |
| 15 | 6, 8 | suplub 9500 | . . . . 5 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧)) |
| 16 | 15 | expdimp 452 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧)) |
| 17 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 18 | brcnvg 5890 | . . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶)) | |
| 19 | 3, 17, 18 | sylancl 586 | . . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶)) |
| 20 | 19 | rexbidv 3179 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
| 21 | 16, 20 | sylibd 239 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
| 22 | 14, 21 | sylbid 240 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
| 23 | 22 | expimpd 453 | 1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 Or wor 5591 ◡ccnv 5684 supcsup 9480 infcinf 9481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-cnv 5693 df-iota 6514 df-riota 7388 df-sup 9482 df-inf 9483 |
| This theorem is referenced by: infnlb 9532 omssubaddlem 34301 omssubadd 34302 gtinf 36320 infxrunb2 45379 |
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