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Theorem cnmpt1plusg 21938
Description: Continuity of the group sum; analogue of cnmpt12f 21517 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 2651 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 21932 . . . . . . . 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 21101 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1366 . . . . . 6 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
10 eqid 2651 . . . . . . 7 (𝑥𝑋𝐴) = (𝑥𝑋𝐴)
1110fmpt 6421 . . . . . 6 (∀𝑥𝑋 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
129, 11sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐴 ∈ (Base‘𝐺))
1312r19.21bi 2961 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
14 cnmpt1plusg.b . . . . . . 7 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
15 cnf2 21101 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
161, 6, 14, 15syl3anc 1366 . . . . . 6 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
17 eqid 2651 . . . . . . 7 (𝑥𝑋𝐵) = (𝑥𝑋𝐵)
1817fmpt 6421 . . . . . 6 (∀𝑥𝑋 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1916, 18sylibr 224 . . . . 5 (𝜑 → ∀𝑥𝑋 𝐵 ∈ (Base‘𝐺))
2019r19.21bi 2961 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
21 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
22 eqid 2651 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
234, 21, 22plusfval 17295 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2413, 20, 23syl2anc 694 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
2524mpteq2dva 4777 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
263, 22tmdcn 21934 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
272, 26syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
281, 7, 14, 27cnmpt12f 21517 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2925, 28eqeltrrd 2731 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  cmpt 4762  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  +𝑓cplusf 17286  TopOnctopon 20763   Cn ccn 21076   ×t ctx 21411  TopMndctmd 21921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-plusf 17288  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-tx 21413  df-tmd 21923
This theorem is referenced by:  tmdmulg  21943  tmdgsum  21946  tmdlactcn  21953  clsnsg  21960  tgpt0  21969  cnmpt1mulr  22032
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