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Theorem cnmpt1plusg 21701
Description: Continuity of the group sum; analogue of cnmpt12f 21279 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpen‘𝐺)
cnmpt1plusg.p + = (+g𝐺)
cnmpt1plusg.g (𝜑𝐺 ∈ TopMnd)
cnmpt1plusg.k (𝜑𝐾 ∈ (TopOn‘𝑋))
cnmpt1plusg.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))
cnmpt1plusg.b (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
Assertion
Ref Expression
cnmpt1plusg (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Distinct variable groups:   𝑥,𝐺   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑋
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   + (𝑥)

Proof of Theorem cnmpt1plusg
StepHypRef Expression
1 cnmpt1plusg.k . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑋))
2 cnmpt1plusg.g . . . . . . . 8 (𝜑𝐺 ∈ TopMnd)
3 tgpcn.j . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
4 eqid 2610 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 21695 . . . . . . . 8 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
62, 5syl 17 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 cnmpt1plusg.a . . . . . . 7 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))
8 cnf2 20863 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
10 eqid 2610 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1110fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
129, 11sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴 ∈ (Base‘𝐺))
1312r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
14 cnmpt1plusg.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
15 cnf2 20863 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
161, 6, 14, 15syl3anc 1318 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
17 eqid 2610 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1817fmpt 6289 . . . . . 6 (∀𝑥𝑋 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1916, 18sylibr 223 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵 ∈ (Base‘𝐺))
2019r19.21bi 2916 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
21 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
22 eqid 2610 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
234, 21, 22plusfval 17071 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2413, 20, 23syl2anc 691 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2524mpteq2dva 4672 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
263, 22tmdcn 21697 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
272, 26syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
281, 7, 14, 27cnmpt12f 21279 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2925, 28eqeltrrd 2689 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  cmpt 4643  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  +𝑓cplusf 17062  TopOnctopon 20518   Cn ccn 20838   ×t ctx 21173  TopMndctmd 21684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-plusf 17064  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-tx 21175  df-tmd 21686
This theorem is referenced by:  tmdmulg  21706  tmdgsum  21709  tmdlactcn  21716  clsnsg  21723  tgpt0  21732  cnmpt1mulr  21795
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