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Theorem cnmpt1plusg 24095
Description: Continuity of the group sum; analogue of cnmpt12f 23674 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 2737 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 24089 . . . . . . 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 23257 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1373 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
109fvmptelcdm 7133 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
11 cnmpt1plusg.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
12 cnf2 23257 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
131, 6, 11, 12syl3anc 1373 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1413fvmptelcdm 7133 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
15 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
16 eqid 2737 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
174, 15, 16plusfval 18660 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1810, 14, 17syl2anc 584 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1918mpteq2dva 5242 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
203, 16tmdcn 24091 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
212, 20syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
221, 7, 11, 21cnmpt12f 23674 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2319, 22eqeltrrd 2842 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  TopOpenctopn 17466  +𝑓cplusf 18650  TopOnctopon 22916   Cn ccn 23232   ×t ctx 23568  TopMndctmd 24078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-plusf 18652  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-tx 23570  df-tmd 24080
This theorem is referenced by:  tmdmulg  24100  tmdgsum  24103  tmdlactcn  24110  clsnsg  24118  tgpt0  24127  cnmpt1mulr  24190
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