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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem3 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 34964. (Contributed by Asger C. Ipsen, 15-Jun-2021.) |
Ref | Expression |
---|---|
knoppndvlem3.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
Ref | Expression |
---|---|
knoppndvlem3 | ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem3.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
2 | elioore 13286 | . . 3 ⊢ (𝐶 ∈ (-1(,)1) → 𝐶 ∈ ℝ) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
4 | eliooord 13315 | . . . 4 ⊢ (𝐶 ∈ (-1(,)1) → (-1 < 𝐶 ∧ 𝐶 < 1)) | |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → (-1 < 𝐶 ∧ 𝐶 < 1)) |
6 | 1red 11152 | . . . 4 ⊢ (𝜑 → 1 ∈ ℝ) | |
7 | 3, 6 | absltd 15306 | . . 3 ⊢ (𝜑 → ((abs‘𝐶) < 1 ↔ (-1 < 𝐶 ∧ 𝐶 < 1))) |
8 | 5, 7 | mpbird 256 | . 2 ⊢ (𝜑 → (abs‘𝐶) < 1) |
9 | 3, 8 | jca 512 | 1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 1c1 11048 < clt 11185 -cneg 11382 (,)cioo 13256 abscabs 15111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-ioo 13260 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 |
This theorem is referenced by: knoppndvlem4 34945 knoppndvlem6 34947 knoppndvlem8 34949 knoppndvlem9 34950 knoppndvlem10 34951 knoppndvlem11 34952 knoppndvlem12 34953 knoppndvlem14 34955 knoppndvlem15 34956 knoppndvlem17 34958 knoppndvlem18 34959 knoppndvlem20 34961 knoppndvlem21 34962 knoppndv 34964 knoppf 34965 knoppcn2 34966 |
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