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Mirrors > Home > ILE Home > Th. List > Mathboxes > dcapnconstALT | GIF version |
Description: Decidability of real number apartness implies the existence of a certain non-constant function from real numbers to integers. A proof of dcapnconst 15475 by means of dceqnconst 15474. (Contributed by Jim Kingdon, 27-Jul-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
dcapnconstALT | ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tridceq 15470 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧 ∨ 𝑧 < 𝑦) → ∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ DECID 𝑦 = 𝑧) | |
2 | reap0 15472 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ (𝑦 < 𝑧 ∨ 𝑦 = 𝑧 ∨ 𝑧 < 𝑦) ↔ ∀𝑥 ∈ ℝ DECID 𝑥 # 0) | |
3 | redc0 15471 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∀𝑧 ∈ ℝ DECID 𝑦 = 𝑧 ↔ ∀𝑥 ∈ ℝ DECID 𝑥 = 0) | |
4 | 1, 2, 3 | 3imtr3i 200 | . 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∀𝑥 ∈ ℝ DECID 𝑥 = 0) |
5 | dceqnconst 15474 | . 2 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 = 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) | |
6 | 4, 5 | syl 14 | 1 ⊢ (∀𝑥 ∈ ℝ DECID 𝑥 # 0 → ∃𝑓(𝑓:ℝ⟶ℤ ∧ (𝑓‘0) = 0 ∧ ∀𝑥 ∈ ℝ+ (𝑓‘𝑥) ≠ 0)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 835 ∨ w3o 979 ∧ w3a 980 = wceq 1364 ∃wex 1503 ≠ wne 2360 ∀wral 2468 class class class wbr 4025 ⟶wf 5238 ‘cfv 5242 ℝcr 7857 0cc0 7858 < clt 8040 # cap 8586 ℤcz 9303 ℝ+crp 9705 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4140 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-un 4458 ax-setind 4561 ax-cnex 7949 ax-resscn 7950 ax-1cn 7951 ax-1re 7952 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-mulrcl 7957 ax-addcom 7958 ax-mulcom 7959 ax-addass 7960 ax-mulass 7961 ax-distr 7962 ax-i2m1 7963 ax-0lt1 7964 ax-1rid 7965 ax-0id 7966 ax-rnegex 7967 ax-precex 7968 ax-cnre 7969 ax-pre-ltirr 7970 ax-pre-ltwlin 7971 ax-pre-lttrn 7972 ax-pre-apti 7973 ax-pre-ltadd 7974 ax-pre-mulgt0 7975 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-csb 3077 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-if 3554 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-int 3867 df-iun 3910 df-br 4026 df-opab 4087 df-mpt 4088 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-rn 4662 df-res 4663 df-ima 4664 df-iota 5203 df-fun 5244 df-fn 5245 df-f 5246 df-f1 5247 df-fo 5248 df-f1o 5249 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-pnf 8042 df-mnf 8043 df-xr 8044 df-ltxr 8045 df-le 8046 df-sub 8178 df-neg 8179 df-reap 8580 df-ap 8587 df-inn 8969 df-z 9304 df-rp 9706 |
This theorem is referenced by: (None) |
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