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Mirrors > Home > ILE Home > Th. List > ivthinclemlm | GIF version |
Description: Lemma for ivthinc 14014. The lower 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 |
---|---|
ivthinclemlm | ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ivth.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | 1 | rexrd 8005 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
3 | ivth.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 3 | rexrd 8005 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
5 | ivth.4 | . . . 4 ⊢ (𝜑 → 𝐴 < 𝐵) | |
6 | 1, 3, 5 | ltled 8074 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
7 | lbicc2 9982 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
8 | 2, 4, 6, 7 | syl3anc 1238 | . 2 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
9 | ivth.9 | . . . 4 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) | |
10 | 9 | simpld 112 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) < 𝑈) |
11 | fveq2 5515 | . . . . 5 ⊢ (𝑤 = 𝐴 → (𝐹‘𝑤) = (𝐹‘𝐴)) | |
12 | 11 | breq1d 4013 | . . . 4 ⊢ (𝑤 = 𝐴 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝐴) < 𝑈)) |
13 | ivthinclem.l | . . . 4 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} | |
14 | 12, 13 | elrab2 2896 | . . 3 ⊢ (𝐴 ∈ 𝐿 ↔ (𝐴 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝐴) < 𝑈)) |
15 | 8, 10, 14 | sylanbrc 417 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐿) |
16 | eleq1 2240 | . . 3 ⊢ (𝑞 = 𝐴 → (𝑞 ∈ 𝐿 ↔ 𝐴 ∈ 𝐿)) | |
17 | 16 | rspcev 2841 | . 2 ⊢ ((𝐴 ∈ (𝐴[,]𝐵) ∧ 𝐴 ∈ 𝐿) → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
18 | 8, 15, 17 | syl2anc 411 | 1 ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 {crab 2459 ⊆ wss 3129 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 ℝ*cxr 7989 < clt 7990 ≤ cle 7991 [,]cicc 9889 –cn→ccncf 13950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltirr 7922 ax-pre-lttrn 7924 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-icc 9893 |
This theorem is referenced by: ivthinclemex 14013 |
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