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Theorem cnmpt1plusg 22695
Description: Continuity of the group sum; analogue of cnmpt12f 22274 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 . . . . . 6 (𝜑𝐾 ∈ (TopOn‘𝑋))
2 cnmpt1plusg.g . . . . . . 7 (𝜑𝐺 ∈ TopMnd)
3 tgpcn.j . . . . . . . 8 𝐽 = (TopOpen‘𝐺)
4 eqid 2824 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 22689 . . . . . . 7 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
62, 5syl 17 . . . . . 6 (𝜑𝐽 ∈ (TopOn‘(Base‘𝐺)))
7 cnmpt1plusg.a . . . . . 6 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))
8 cnf2 21857 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
109fvmptelrn 6868 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
11 cnmpt1plusg.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
12 cnf2 21857 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
131, 6, 11, 12syl3anc 1368 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1413fvmptelrn 6868 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
15 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
16 eqid 2824 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
174, 15, 16plusfval 17859 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1810, 14, 17syl2anc 587 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1918mpteq2dva 5147 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
203, 16tmdcn 22691 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
212, 20syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
221, 7, 11, 21cnmpt12f 22274 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2319, 22eqeltrrd 2917 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cmpt 5132  wf 6339  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  TopOpenctopn 16695  +𝑓cplusf 17849  TopOnctopon 21518   Cn ccn 21832   ×t ctx 22168  TopMndctmd 22678
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-topgen 16717  df-plusf 17851  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-tx 22170  df-tmd 22680
This theorem is referenced by:  tmdmulg  22700  tmdgsum  22703  tmdlactcn  22710  clsnsg  22718  tgpt0  22727  cnmpt1mulr  22790
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