Theorem cnmpt1plusg 24209

Description: Continuity of the group sum; analogue of cnmpt12f 23788 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 2769	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	3, 4	tmdtopon 24203	. . . . . . 7 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
6	2, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
7		cnmpt1plusg.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
8		cnf2 23371	. . . . . 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 1396	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺))
10	9	fvmptelcdm 7106	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺))
11		cnmpt1plusg.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))
12		cnf2 23371	. . . . . 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 1396	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺))
14	13	fvmptelcdm 7106	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺))
15		cnmpt1plusg.p	. . . . 5 ⊢ + = (+g‘𝐺)
16		eqid 2769	. . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
17	4, 15, 16	plusfval 18701	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
18	10, 14, 17	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
19	18	mpteq2dva 5205	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)))
20	3, 16	tmdcn 24205	. . . 4 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
21	2, 20	syl 18	. . 3 ⊢ (𝜑 → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
22	1, 7, 11, 21	cnmpt12f 23788	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
23	19, 22	eqeltrrd 2870	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ↦ cmpt 5193  ⟶wf 6529  ‘cfv 6533  (class class class)co 7408  Basecbs 17265  +_gcplusg 17306  TopOpenctopn 17470  +_𝑓cplusf 18691  TopOnctopon 23032   Cn ccn 23346   ×_t ctx 23682  TopMndctmd 24192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-map 8822  df-topgen 17492  df-plusf 18693  df-top 23016  df-topon 23033  df-topsp 23055  df-bases 23068  df-cn 23349  df-tx 23684  df-tmd 24194

This theorem is referenced by:  tmdmulg  24214  tmdgsum  24217  tmdlactcn  24224  clsnsg  24232  tgpt0  24241  cnmpt1mulr  24304