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Mirrors > Home > ILE Home > Th. List > ivthinclemex | GIF version |
Description: Lemma for ivthinc 13963. Existence of a number between the lower cut and the upper cut. (Contributed by Jim Kingdon, 20-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 |
---|---|
ivthinclemex | ⊢ (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑧 ∧ ∀𝑟 ∈ 𝑅 𝑧 < 𝑟)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ivth.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ivth.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ivthinclem.l | . . . 4 ⊢ 𝐿 = {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} | |
4 | ssrab2 3240 | . . . 4 ⊢ {𝑤 ∈ (𝐴[,]𝐵) ∣ (𝐹‘𝑤) < 𝑈} ⊆ (𝐴[,]𝐵) | |
5 | 3, 4 | eqsstri 3187 | . . 3 ⊢ 𝐿 ⊆ (𝐴[,]𝐵) |
6 | 5 | a1i 9 | . 2 ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) |
7 | ivthinclem.r | . . . 4 ⊢ 𝑅 = {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} | |
8 | ssrab2 3240 | . . . 4 ⊢ {𝑤 ∈ (𝐴[,]𝐵) ∣ 𝑈 < (𝐹‘𝑤)} ⊆ (𝐴[,]𝐵) | |
9 | 7, 8 | eqsstri 3187 | . . 3 ⊢ 𝑅 ⊆ (𝐴[,]𝐵) |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → 𝑅 ⊆ (𝐴[,]𝐵)) |
11 | ivth.3 | . . 3 ⊢ (𝜑 → 𝑈 ∈ ℝ) | |
12 | ivth.4 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐵) | |
13 | ivth.5 | . . 3 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ 𝐷) | |
14 | ivth.7 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐷–cn→ℂ)) | |
15 | ivth.8 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (𝐹‘𝑥) ∈ ℝ) | |
16 | ivth.9 | . . 3 ⊢ (𝜑 → ((𝐹‘𝐴) < 𝑈 ∧ 𝑈 < (𝐹‘𝐵))) | |
17 | ivthinc.i | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴[,]𝐵)) ∧ (𝑦 ∈ (𝐴[,]𝐵) ∧ 𝑥 < 𝑦)) → (𝐹‘𝑥) < (𝐹‘𝑦)) | |
18 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemlm 13954 | . 2 ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
19 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemum 13955 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑅) |
20 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemlr 13957 | . 2 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
21 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemur 13959 | . 2 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑅 ↔ ∃𝑞 ∈ 𝑅 𝑞 < 𝑟)) |
22 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemdisj 13960 | . 2 ⊢ (𝜑 → (𝐿 ∩ 𝑅) = ∅) |
23 | 1, 2, 11, 12, 13, 14, 15, 16, 17, 3, 7 | ivthinclemloc 13961 | . 2 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑅))) |
24 | 1, 2, 6, 10, 18, 19, 20, 21, 22, 23, 12 | dedekindicc 13953 | 1 ⊢ (𝜑 → ∃!𝑧 ∈ (𝐴(,)𝐵)(∀𝑞 ∈ 𝐿 𝑞 < 𝑧 ∧ ∀𝑟 ∈ 𝑅 𝑧 < 𝑟)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃!wreu 2457 {crab 2459 ⊆ wss 3129 class class class wbr 4001 ‘cfv 5213 (class class class)co 5870 ℂcc 7804 ℝcr 7805 < clt 7986 (,)cioo 9882 [,]cicc 9885 –cn→ccncf 13899 |
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-coll 4116 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-iinf 4585 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 ax-pre-mulext 7924 ax-arch 7925 ax-caucvg 7926 ax-pre-suploc 7927 |
This theorem depends on definitions: df-bi 117 df-dc 835 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-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-tr 4100 df-id 4291 df-po 4294 df-iso 4295 df-iord 4364 df-on 4366 df-ilim 4367 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-isom 5222 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-1st 6136 df-2nd 6137 df-recs 6301 df-frec 6387 df-map 6645 df-sup 6978 df-inf 6979 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 df-div 8624 df-inn 8914 df-2 8972 df-3 8973 df-4 8974 df-n0 9171 df-z 9248 df-uz 9523 df-rp 9648 df-ioo 9886 df-icc 9889 df-seqfrec 10439 df-exp 10513 df-cj 10842 df-re 10843 df-im 10844 df-rsqrt 10998 df-abs 10999 df-cncf 13900 |
This theorem is referenced by: ivthinc 13963 |
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