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Mirrors > Home > ILE Home > Th. List > ivthinclemum | GIF version |
Description: Lemma for ivthinc 12793. The upper cut is bounded. (Contributed by Jim Kingdon, 18-Feb-2024.) |
Ref | Expression |
---|---|
ivth.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ivth.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ivth.3 | ⊢ (𝜑 → 𝑈 ∈ ℝ) |
ivth.4 | ⊢ (𝜑 → 𝐴 < 𝐵) |
ivth.5 | ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) |
ivth.7 | ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) |
ivth.8 | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) |
ivth.9 | ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) |
ivthinc.i | ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦)) |
ivthinclem.l | ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} |
ivthinclem.r | ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} |
Ref | Expression |
---|---|
ivthinclemum | ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ivth.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 7818 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | ivth.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 7818 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | ivth.4 | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐵) | |
6 | 1, 3, 5 | ltled 7884 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | ubicc2 9771 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) | |
8 | 2, 4, 6, 7 | syl3anc 1216 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵)) |
9 | ivth.9 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) | |
10 | 9 | simprd 113 | . . 3 ⊢ (𝜑 → 𝑈 < (𝐹‘𝐵)) |
11 | fveq2 5421 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐹‘𝑤) = (𝐹‘𝐵)) | |
12 | 11 | breq2d 3941 | . . . 4 ⊢ (𝑤 = 𝐵 → (𝑈 < (𝐹‘𝑤) ↔ 𝑈 < (𝐹‘𝐵))) |
13 | ivthinclem.r | . . . 4 ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} | |
14 | 12, 13 | elrab2 2843 | . . 3 ⊢ (𝐵 ∈ 𝑅 ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝐵))) |
15 | 8, 10, 14 | sylanbrc 413 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑅) |
16 | eleq1 2202 | . . 3 ⊢ (𝑟 = 𝐵 → (𝑟 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅)) | |
17 | 16 | rspcev 2789 | . 2 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ 𝑅) → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
18 | 8, 15, 17 | syl2anc 408 | 1 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 {crab 2420 ⊆ wss 3071 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7621 ℝcr 7622 ℝ*cxr 7802 < clt 7803 ≤ cle 7804 [,]cicc 9677 –cn→ccncf 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-pre-ltirr 7735 ax-pre-lttrn 7737 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-icc 9681 |
This theorem is referenced by: ivthinclemex 12792 |
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