Theorem cnmpt1plusg 24111

Description: Continuity of the group sum; analogue of cnmpt12f 23690 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	⊢ + = (+g‘𝐺)
cnmpt1plusg.g	⊢ (𝜑 → 𝐺 ∈ TopMnd)
cnmpt1plusg.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
cnmpt1plusg.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
cnmpt1plusg.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))

Assertion

Ref	Expression
cnmpt1plusg	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (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

Step	Hyp	Ref	Expression
1		cnmpt1plusg.k	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
2		cnmpt1plusg.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
3		tgpcn.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝐺)
4		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	3, 4	tmdtopon 24105	. . . . . . 7 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		cnmpt1plusg.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
8		cnf2 23273	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺))
9	1, 6, 7, 8	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺))
10	9	fvmptelcdm 7133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺))
11		cnmpt1plusg.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))
12		cnf2 23273	. . . . . 6 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺))
13	1, 6, 11, 12	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺))
14	13	fvmptelcdm 7133	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺))
15		cnmpt1plusg.p	. . . . 5 ⊢ + = (+g‘𝐺)
16		eqid 2735	. . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
17	4, 15, 16	plusfval 18673	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
18	10, 14, 17	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
19	18	mpteq2dva 5248	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)))
20	3, 16	tmdcn 24107	. . . 4 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
21	2, 20	syl 17	. . 3 ⊢ (𝜑 → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
22	1, 7, 11, 21	cnmpt12f 23690	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
23	19, 22	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  TopOpenctopn 17468  +_𝑓cplusf 18663  TopOnctopon 22932   Cn ccn 23248   ×_t ctx 23584  TopMndctmd 24094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-plusf 18665  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-tx 23586  df-tmd 24096

This theorem is referenced by:  tmdmulg  24116  tmdgsum  24119  tmdlactcn  24126  clsnsg  24134  tgpt0  24143  cnmpt1mulr  24206