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Mirrors > Home > ILE Home > Th. List > ivthinclemdisj | GIF version |
Description: Lemma for ivthinc 14052. The lower and upper cuts are disjoint. (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 |
---|---|
ivthinclemdisj | ⊢ (𝜑 → (𝐿 ∩ 𝑅) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5515 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
2 | 1 | eleq1d 2246 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑧) ∈ ℝ)) |
3 | ivth.8 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
4 | 3 | ralrimiva 2550 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
5 | 4 | adantr 276 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
6 | fveq2 5515 | . . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧)) | |
7 | 6 | breq1d 4013 | . . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑧) < 𝑈)) |
8 | ivthinclem.l | . . . . . . . . . . 11 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} | |
9 | 7, 8 | elrab2 2896 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐿 ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
10 | 9 | biimpi 120 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝐿 → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
11 | 10 | adantl 277 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
12 | 11 | simpld 112 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → 𝑧 ∈ (𝐴[,]𝐵)) |
13 | 2, 5, 12 | rspcdva 2846 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝐹‘𝑧) ∈ ℝ) |
14 | ivth.3 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ) | |
15 | 14 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → 𝑈 ∈ ℝ) |
16 | 11 | simprd 114 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝐹‘𝑧) < 𝑈) |
17 | 13, 15, 16 | ltnsymd 8075 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ 𝑈 < (𝐹‘𝑧)) |
18 | 17 | intnand 931 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ (𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝑧))) |
19 | 6 | breq2d 4015 | . . . . 5 ⊢ (𝑤 = 𝑧 → (𝑈 < (𝐹‘𝑤) ↔ 𝑈 < (𝐹‘𝑧))) |
20 | ivthinclem.r | . . . . 5 ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} | |
21 | 19, 20 | elrab2 2896 | . . . 4 ⊢ (𝑧 ∈ 𝑅 ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝑧))) |
22 | 18, 21 | sylnibr 677 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ 𝑧 ∈ 𝑅) |
23 | 22 | ralrimiva 2550 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐿 ¬ 𝑧 ∈ 𝑅) |
24 | disj 3471 | . 2 ⊢ ((𝐿 ∩ 𝑅) = ∅ ↔ ∀𝑧 ∈ 𝐿 ¬ 𝑧 ∈ 𝑅) | |
25 | 23, 24 | sylibr 134 | 1 ⊢ (𝜑 → (𝐿 ∩ 𝑅) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 ∩ cin 3128 ⊆ wss 3129 ∅c0 3422 class class class wbr 4003 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 ℝcr 7809 < clt 7990 [,]cicc 9889 –cn→ccncf 13988 |
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-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-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-iota 5178 df-fv 5224 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 |
This theorem is referenced by: ivthinclemex 14051 |
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