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Theorem dceqnconst 15193
Description: Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 15188 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.)
Assertion
Ref Expression
dceqnconst (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ βˆƒπ‘“(𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0))
Distinct variable group:   π‘₯,𝑓

Proof of Theorem dceqnconst
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 7963 . . . 4 ℝ ∈ V
21mptex 5758 . . 3 (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) ∈ V
32a1i 9 . 2 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) ∈ V)
4 0zd 9283 . . . . 5 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑦 ∈ ℝ) β†’ 0 ∈ β„€)
5 1zzd 9298 . . . . 5 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„€)
6 eqeq1 2196 . . . . . . 7 (π‘₯ = 𝑦 β†’ (π‘₯ = 0 ↔ 𝑦 = 0))
76dcbid 839 . . . . . 6 (π‘₯ = 𝑦 β†’ (DECID π‘₯ = 0 ↔ DECID 𝑦 = 0))
87rspccva 2855 . . . . 5 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑦 ∈ ℝ) β†’ DECID 𝑦 = 0)
94, 5, 8ifcldcd 3585 . . . 4 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑦 ∈ ℝ) β†’ if(𝑦 = 0, 0, 1) ∈ β„€)
109fmpttd 5687 . . 3 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):β„βŸΆβ„€)
11 0re 7975 . . . . . 6 0 ∈ ℝ
12 0zd 9283 . . . . . . . 8 (⊀ β†’ 0 ∈ β„€)
13 1zzd 9298 . . . . . . . 8 (⊀ β†’ 1 ∈ β„€)
14 eqid 2189 . . . . . . . . . . 11 0 = 0
1514orci 732 . . . . . . . . . 10 (0 = 0 ∨ Β¬ 0 = 0)
16 df-dc 836 . . . . . . . . . 10 (DECID 0 = 0 ↔ (0 = 0 ∨ Β¬ 0 = 0))
1715, 16mpbir 146 . . . . . . . . 9 DECID 0 = 0
1817a1i 9 . . . . . . . 8 (⊀ β†’ DECID 0 = 0)
1912, 13, 18ifcldcd 3585 . . . . . . 7 (⊀ β†’ if(0 = 0, 0, 1) ∈ β„€)
2019mptru 1373 . . . . . 6 if(0 = 0, 0, 1) ∈ β„€
21 eqeq1 2196 . . . . . . . 8 (𝑦 = 0 β†’ (𝑦 = 0 ↔ 0 = 0))
2221ifbid 3570 . . . . . . 7 (𝑦 = 0 β†’ if(𝑦 = 0, 0, 1) = if(0 = 0, 0, 1))
23 eqid 2189 . . . . . . 7 (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))
2422, 23fvmptg 5608 . . . . . 6 ((0 ∈ ℝ ∧ if(0 = 0, 0, 1) ∈ β„€) β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = if(0 = 0, 0, 1))
2511, 20, 24mp2an 426 . . . . 5 ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = if(0 = 0, 0, 1)
2614iftruei 3555 . . . . 5 if(0 = 0, 0, 1) = 0
2725, 26eqtri 2210 . . . 4 ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = 0
2827a1i 9 . . 3 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = 0)
29 1ne0 9005 . . . . . 6 1 β‰  0
30 eqeq1 2196 . . . . . . . . . 10 (𝑦 = 𝑧 β†’ (𝑦 = 0 ↔ 𝑧 = 0))
3130ifbid 3570 . . . . . . . . 9 (𝑦 = 𝑧 β†’ if(𝑦 = 0, 0, 1) = if(𝑧 = 0, 0, 1))
32 rpre 9678 . . . . . . . . . 10 (𝑧 ∈ ℝ+ β†’ 𝑧 ∈ ℝ)
3332adantl 277 . . . . . . . . 9 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ 𝑧 ∈ ℝ)
34 0zd 9283 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ 0 ∈ β„€)
35 1zzd 9298 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ 1 ∈ β„€)
36 eqeq1 2196 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ (π‘₯ = 0 ↔ 𝑧 = 0))
3736dcbid 839 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ (DECID π‘₯ = 0 ↔ DECID 𝑧 = 0))
38 simpl 109 . . . . . . . . . . 11 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0)
3937, 38, 33rspcdva 2861 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ DECID 𝑧 = 0)
4034, 35, 39ifcldcd 3585 . . . . . . . . 9 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ if(𝑧 = 0, 0, 1) ∈ β„€)
4123, 31, 33, 40fvmptd3 5625 . . . . . . . 8 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) = if(𝑧 = 0, 0, 1))
42 rpne0 9687 . . . . . . . . . . 11 (𝑧 ∈ ℝ+ β†’ 𝑧 β‰  0)
4342neneqd 2381 . . . . . . . . . 10 (𝑧 ∈ ℝ+ β†’ Β¬ 𝑧 = 0)
4443iffalsed 3559 . . . . . . . . 9 (𝑧 ∈ ℝ+ β†’ if(𝑧 = 0, 0, 1) = 1)
4544adantl 277 . . . . . . . 8 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ if(𝑧 = 0, 0, 1) = 1)
4641, 45eqtrd 2222 . . . . . . 7 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) = 1)
4746neeq1d 2378 . . . . . 6 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) β‰  0 ↔ 1 β‰  0))
4829, 47mpbiri 168 . . . . 5 ((βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 ∧ 𝑧 ∈ ℝ+) β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) β‰  0)
4948ralrimiva 2563 . . . 4 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ βˆ€π‘§ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) β‰  0)
50 fveq2 5530 . . . . . 6 (𝑧 = π‘₯ β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯))
5150neeq1d 2378 . . . . 5 (𝑧 = π‘₯ β†’ (((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) β‰  0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0))
5251cbvralv 2718 . . . 4 (βˆ€π‘§ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘§) β‰  0 ↔ βˆ€π‘₯ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0)
5349, 52sylib 122 . . 3 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ βˆ€π‘₯ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0)
5410, 28, 533jca 1179 . 2 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):β„βŸΆβ„€ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0))
55 feq1 5363 . . 3 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ (𝑓:β„βŸΆβ„€ ↔ (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):β„βŸΆβ„€))
56 fveq1 5529 . . . 4 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ (π‘“β€˜0) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0))
5756eqeq1d 2198 . . 3 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ ((π‘“β€˜0) = 0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = 0))
58 fveq1 5529 . . . . 5 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ (π‘“β€˜π‘₯) = ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯))
5958neeq1d 2378 . . . 4 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ ((π‘“β€˜π‘₯) β‰  0 ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0))
6059ralbidv 2490 . . 3 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ (βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0 ↔ βˆ€π‘₯ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0))
6155, 57, 603anbi123d 1323 . 2 (𝑓 = (𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)) β†’ ((𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0) ↔ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1)):β„βŸΆβ„€ ∧ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ ((𝑦 ∈ ℝ ↦ if(𝑦 = 0, 0, 1))β€˜π‘₯) β‰  0)))
623, 54, 61elabd 2897 1 (βˆ€π‘₯ ∈ ℝ DECID π‘₯ = 0 β†’ βˆƒπ‘“(𝑓:β„βŸΆβ„€ ∧ (π‘“β€˜0) = 0 ∧ βˆ€π‘₯ ∈ ℝ+ (π‘“β€˜π‘₯) β‰  0))
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   ∧ w3a 980   = wceq 1364  βŠ€wtru 1365  βˆƒwex 1503   ∈ wcel 2160   β‰  wne 2360  βˆ€wral 2468  Vcvv 2752  ifcif 3549   ↦ cmpt 4079  βŸΆwf 5227  β€˜cfv 5231  β„cr 7828  0cc0 7829  1c1 7830  β„€cz 9271  β„+crp 9671
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 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-addcom 7929  ax-addass 7931  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-0id 7937  ax-rnegex 7938  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-ltadd 7945
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 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-inn 8938  df-z 9272  df-rp 9672
This theorem is referenced by:  dcapnconstALT  15195
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