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Mirrors > Home > ILE Home > Th. List > ivthinclemum | GIF version |
Description: Lemma for ivthinc 13162. 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 7939 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | ivth.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 7939 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | ivth.4 | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐵) | |
6 | 1, 3, 5 | ltled 8008 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | ubicc2 9912 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) | |
8 | 2, 4, 6, 7 | syl3anc 1227 | . 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 5480 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐹‘𝑤) = (𝐹‘𝐵)) | |
12 | 11 | breq2d 3988 | . . . 4 ⊢ (𝑤 = 𝐵 → (𝑈 < (𝐹‘𝑤) ↔ 𝑈 < (𝐹‘𝐵))) |
13 | ivthinclem.r | . . . 4 ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} | |
14 | 12, 13 | elrab2 2880 | . . 3 ⊢ (𝐵 ∈ 𝑅 ↔ (𝐵 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝐵))) |
15 | 8, 10, 14 | sylanbrc 414 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑅) |
16 | eleq1 2227 | . . 3 ⊢ (𝑟 = 𝐵 → (𝑟 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅)) | |
17 | 16 | rspcev 2825 | . 2 ⊢ ((𝐵 ∈ (𝐴[,]𝐵) ∧ 𝐵 ∈ 𝑅) → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
18 | 8, 15, 17 | syl2anc 409 | 1 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 {crab 2446 ⊆ wss 3111 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 ℝ*cxr 7923 < clt 7924 ≤ cle 7925 [,]cicc 9818 –cn→ccncf 13098 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-lttrn 7858 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-icc 9822 |
This theorem is referenced by: ivthinclemex 13161 |
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