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Theorem cnmpt1plusg 21796
 Description: Continuity of the group sum; analogue of cnmpt12f 21374 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 2626 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 21790 . . . . . . . 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 20958 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1323 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
10 eqid 2626 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1110fmpt 6338 . . . . . 6 (∀𝑥𝑋 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
129, 11sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴 ∈ (Base‘𝐺))
1312r19.21bi 2932 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
14 cnmpt1plusg.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
15 cnf2 20958 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
161, 6, 14, 15syl3anc 1323 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
17 eqid 2626 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1817fmpt 6338 . . . . . 6 (∀𝑥𝑋 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1916, 18sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵 ∈ (Base‘𝐺))
2019r19.21bi 2932 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
21 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
22 eqid 2626 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
234, 21, 22plusfval 17164 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2413, 20, 23syl2anc 692 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2524mpteq2dva 4709 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
263, 22tmdcn 21792 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
272, 26syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
281, 7, 14, 27cnmpt12f 21374 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2925, 28eqeltrrd 2705 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1992  ∀wral 2912   ↦ cmpt 4678  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  TopOpenctopn 15998  +𝑓cplusf 17155  TopOnctopon 20613   Cn ccn 20933   ×t ctx 21268  TopMndctmd 21779 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-topgen 16020  df-plusf 17157  df-top 20616  df-bases 20617  df-topon 20618  df-topsp 20619  df-cn 20936  df-tx 21270  df-tmd 21781 This theorem is referenced by:  tmdmulg  21801  tmdgsum  21804  tmdlactcn  21811  clsnsg  21818  tgpt0  21827  cnmpt1mulr  21890
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