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Theorem cnmpt1plusg 22389
Description: Continuity of the group sum; analogue of cnmpt12f 21968 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 2772 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
53, 4tmdtopon 22383 . . . . . . 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 21551 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
91, 6, 7, 8syl3anc 1351 . . . . 5 (𝜑 → (𝑥𝑋𝐴):𝑋⟶(Base‘𝐺))
109fvmptelrn 6694 . . . 4 ((𝜑𝑥𝑋) → 𝐴 ∈ (Base‘𝐺))
11 cnmpt1plusg.b . . . . . 6 (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))
12 cnf2 21551 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
131, 6, 11, 12syl3anc 1351 . . . . 5 (𝜑 → (𝑥𝑋𝐵):𝑋⟶(Base‘𝐺))
1413fvmptelrn 6694 . . . 4 ((𝜑𝑥𝑋) → 𝐵 ∈ (Base‘𝐺))
15 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
16 eqid 2772 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
174, 15, 16plusfval 17706 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1810, 14, 17syl2anc 576 . . 3 ((𝜑𝑥𝑋) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
1918mpteq2dva 5016 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋 ↦ (𝐴 + 𝐵)))
203, 16tmdcn 22385 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
212, 20syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
221, 7, 11, 21cnmpt12f 21968 . 2 (𝜑 → (𝑥𝑋 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
2319, 22eqeltrrd 2861 1 (𝜑 → (𝑥𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  cmpt 5002  wf 6178  cfv 6182  (class class class)co 6970  Basecbs 16329  +gcplusg 16411  TopOpenctopn 16541  +𝑓cplusf 17697  TopOnctopon 21212   Cn ccn 21526   ×t ctx 21862  TopMndctmd 22372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-map 8200  df-topgen 16563  df-plusf 17699  df-top 21196  df-topon 21213  df-topsp 21235  df-bases 21248  df-cn 21529  df-tx 21864  df-tmd 22374
This theorem is referenced by:  tmdmulg  22394  tmdgsum  22397  tmdlactcn  22404  clsnsg  22411  tgpt0  22420  cnmpt1mulr  22483
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