Theorem cnmpt2plusg 22269

Description: Continuity of the group sum; analogue of cnmpt22f 21856 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))
cnmpt2plusg.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))
cnmpt2plusg.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
cnmpt2plusg.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))

Assertion

Ref	Expression
cnmpt2plusg	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝐽,𝑦   𝑥,𝐾   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   + (𝑥,𝑦)   𝐾(𝑦)   𝐿(𝑥,𝑦)

Proof of Theorem cnmpt2plusg

Step	Hyp	Ref	Expression
1		cnmpt1plusg.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋))
2		cnmpt2plusg.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌))
3		txtopon 21772	. . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
5		cnmpt1plusg.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TopMnd)
6		tgpcn.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝐺)
7		eqid 2825	. . . . . . . . . . 11 ⊢ (Base‘𝐺) = (Base‘𝐺)
8	6, 7	tmdtopon 22262	. . . . . . . . . 10 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
9	5, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
10		cnmpt2plusg.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
11		cnf2 21431	. . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
12	4, 9, 10, 11	syl3anc 1494	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
13		eqid 2825	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
14	13	fmpt2 7505	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
15	12, 14	sylibr 226	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺))
16	15	r19.21bi 3141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐺))
17	16	r19.21bi 3141	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ (Base‘𝐺))
18	17	3impa 1140	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ (Base‘𝐺))
19		cnmpt2plusg.b	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
20		cnf2 21431	. . . . . . . . 9 ⊢ (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
21	4, 9, 19, 20	syl3anc 1494	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
22		eqid 2825	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
23	22	fmpt2 7505	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
24	21, 23	sylibr 226	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺))
25	24	r19.21bi 3141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝐺))
26	25	r19.21bi 3141	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ (Base‘𝐺))
27	26	3impa 1140	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ (Base‘𝐺))
28		cnmpt1plusg.p	. . . . 5 ⊢ + = (+g‘𝐺)
29		eqid 2825	. . . . 5 ⊢ (+𝑓‘𝐺) = (+𝑓‘𝐺)
30	7, 28, 29	plusfval 17608	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
31	18, 27, 30	syl2anc 579	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴(+𝑓‘𝐺)𝐵) = (𝐴 + 𝐵))
32	31	mpt2eq3dva 6984	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴(+𝑓‘𝐺)𝐵)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 + 𝐵)))
33	6, 29	tmdcn 22264	. . . 4 ⊢ (𝐺 ∈ TopMnd → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
34	5, 33	syl 17	. . 3 ⊢ (𝜑 → (+𝑓‘𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
35	1, 2, 10, 19, 34	cnmpt22f 21856	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴(+𝑓‘𝐺)𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
36	32, 35	eqeltrrd 2907	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   × cxp 5344  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910   ↦ cmpt2 6912  Basecbs 16229  +_gcplusg 16312  TopOpenctopn 16442  +_𝑓cplusf 17599  TopOnctopon 21092   Cn ccn 21406   ×_t ctx 21741  TopMndctmd 22251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-topgen 16464  df-plusf 17601  df-top 21076  df-topon 21093  df-topsp 21115  df-bases 21128  df-cn 21409  df-tx 21743  df-tmd 22253

This theorem is referenced by:  tgpsubcn  22271  oppgtmd  22278  prdstmdd  22304  cnmpt2mulr  22363