Theorem cnmpt1plusg 24116

Description: Continuity of the group sum; analogue of cnmpt12f 23695 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 2740	. . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺)
5	3, 4	tmdtopon 24110	. . . . . . 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 23278	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐺))
10	9	fvmptelcdm 7147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐺))
11		cnmpt1plusg.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽))
12		cnf2 23278	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝐺))
14	13	fvmptelcdm 7147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝐺))
15		cnmpt1plusg.p	. . . . 5 ⊢ + = (+g‘𝐺)
16		eqid 2740	. . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
17	4, 15, 16	plusfval 18685	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
18	10, 14, 17	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
19	18	mpteq2dva 5266	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)))
20	3, 16	tmdcn 24112	. . . 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 23695	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ (𝐾 Cn 𝐽))
23	19, 22	eqeltrrd 2845	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 + 𝐵)) ∈ (𝐾 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  TopOpenctopn 17481  +_𝑓cplusf 18675  TopOnctopon 22937   Cn ccn 23253   ×_t ctx 23589  TopMndctmd 24099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-topgen 17503  df-plusf 18677  df-top 22921  df-topon 22938  df-topsp 22960  df-bases 22974  df-cn 23256  df-tx 23591  df-tmd 24101

This theorem is referenced by:  tmdmulg  24121  tmdgsum  24124  tmdlactcn  24131  clsnsg  24139  tgpt0  24148  cnmpt1mulr  24211