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Theorem cncfmptid 12789
 Description: The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
Assertion
Ref Expression
cncfmptid ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
Distinct variable groups:   𝑥,𝑆   𝑥,𝑇

Proof of Theorem cncfmptid
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstr 3109 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑆 ⊆ ℂ)
2 simpr 109 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → 𝑇 ⊆ ℂ)
3 simpll 519 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑆𝑇)
4 simpr 109 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑆)
53, 4sseldd 3102 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ 𝑥𝑆) → 𝑥𝑇)
65fmpttd 5582 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥):𝑆𝑇)
7 simpr 109 . . . 4 ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+)
87a1i 9 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑦𝑆𝑤 ∈ ℝ+) → 𝑤 ∈ ℝ+))
9 eqid 2140 . . . . . . . 8 (𝑥𝑆𝑥) = (𝑥𝑆𝑥)
10 id 19 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
11 simprll 527 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑦𝑆)
129, 10, 11, 11fvmptd3 5521 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑦) = 𝑦)
13 id 19 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
14 simprlr 528 . . . . . . . 8 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → 𝑧𝑆)
159, 13, 14, 14fvmptd3 5521 . . . . . . 7 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((𝑥𝑆𝑥)‘𝑧) = 𝑧)
1612, 15oveq12d 5799 . . . . . 6 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧)) = (𝑦𝑧))
1716fveq2d 5432 . . . . 5 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) = (abs‘(𝑦𝑧)))
1817breq1d 3946 . . . 4 (((𝑆𝑇𝑇 ⊆ ℂ) ∧ ((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+)) → ((abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤 ↔ (abs‘(𝑦𝑧)) < 𝑤))
1918exbiri 380 . . 3 ((𝑆𝑇𝑇 ⊆ ℂ) → (((𝑦𝑆𝑧𝑆) ∧ 𝑤 ∈ ℝ+) → ((abs‘(𝑦𝑧)) < 𝑤 → (abs‘(((𝑥𝑆𝑥)‘𝑦) − ((𝑥𝑆𝑥)‘𝑧))) < 𝑤)))
206, 8, 19elcncf1di 12772 . 2 ((𝑆𝑇𝑇 ⊆ ℂ) → ((𝑆 ⊆ ℂ ∧ 𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇)))
211, 2, 20mp2and 430 1 ((𝑆𝑇𝑇 ⊆ ℂ) → (𝑥𝑆𝑥) ∈ (𝑆cn𝑇))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 1481   ⊆ wss 3075   class class class wbr 3936   ↦ cmpt 3996  ‘cfv 5130  (class class class)co 5781  ℂcc 7641   < clt 7823   − cmin 7956  ℝ+crp 9469  abscabs 10800  –cn→ccncf 12763 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-map 6551  df-cncf 12764 This theorem is referenced by:  expcncf  12798  dvcnp2cntop  12869
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