Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ivthinclemdisj | GIF version |
Description: Lemma for ivthinc 12793. 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 5421 | . . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
2 | 1 | eleq1d 2208 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ∈ ℝ ↔ (𝐹‘𝑧) ∈ ℝ)) |
3 | ivth.8 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
4 | 3 | ralrimiva 2505 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
5 | 4 | adantr 274 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ∀𝑥 ∈ (𝐴[,]𝐵)(𝐹‘𝑥) ∈ ℝ) |
6 | fveq2 5421 | . . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → (𝐹‘𝑤) = (𝐹‘𝑧)) | |
7 | 6 | breq1d 3939 | . . . . . . . . . . 11 ⊢ (𝑤 = 𝑧 → ((𝐹‘𝑤) < 𝑈 ↔ (𝐹‘𝑧) < 𝑈)) |
8 | ivthinclem.l | . . . . . . . . . . 11 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} | |
9 | 7, 8 | elrab2 2843 | . . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐿 ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
10 | 9 | biimpi 119 | . . . . . . . . 9 ⊢ (𝑧 ∈ 𝐿 → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
11 | 10 | adantl 275 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝑧 ∈ (𝐴[,]𝐵) ∧ (𝐹‘𝑧) < 𝑈)) |
12 | 11 | simpld 111 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → 𝑧 ∈ (𝐴[,]𝐵)) |
13 | 2, 5, 12 | rspcdva 2794 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝐹‘𝑧) ∈ ℝ) |
14 | ivth.3 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℝ) | |
15 | 14 | adantr 274 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → 𝑈 ∈ ℝ) |
16 | 11 | simprd 113 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → (𝐹‘𝑧) < 𝑈) |
17 | 13, 15, 16 | ltnsymd 7885 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ 𝑈 < (𝐹‘𝑧)) |
18 | 17 | intnand 916 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ (𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝑧))) |
19 | 6 | breq2d 3941 | . . . . 5 ⊢ (𝑤 = 𝑧 → (𝑈 < (𝐹‘𝑤) ↔ 𝑈 < (𝐹‘𝑧))) |
20 | ivthinclem.r | . . . . 5 ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} | |
21 | 19, 20 | elrab2 2843 | . . . 4 ⊢ (𝑧 ∈ 𝑅 ↔ (𝑧 ∈ (𝐴[,]𝐵) ∧ 𝑈 < (𝐹‘𝑧))) |
22 | 18, 21 | sylnibr 666 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐿) → ¬ 𝑧 ∈ 𝑅) |
23 | 22 | ralrimiva 2505 | . 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐿 ¬ 𝑧 ∈ 𝑅) |
24 | disj 3411 | . 2 ⊢ ((𝐿 ∩ 𝑅) = ∅ ↔ ∀𝑧 ∈ 𝐿 ¬ 𝑧 ∈ 𝑅) | |
25 | 23, 24 | sylibr 133 | 1 ⊢ (𝜑 → (𝐿 ∩ 𝑅) = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 {crab 2420 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ℂcc 7621 ℝcr 7622 < clt 7803 [,]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-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-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-iota 5088 df-fv 5131 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 |
This theorem is referenced by: ivthinclemex 12792 |
Copyright terms: Public domain | W3C validator |