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Theorem cnmpt2plusg 21802
Description: Continuity of the group sum; analogue of cnmpt22f 21388 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))
cnmpt2plusg.l (𝜑𝐿 ∈ (TopOn‘𝑌))
cnmpt2plusg.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
cnmpt2plusg.b (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Assertion
Ref Expression
cnmpt2plusg (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (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
StepHypRef Expression
1 cnmpt1plusg.k . . . . . . . . . 10 (𝜑𝐾 ∈ (TopOn‘𝑋))
2 cnmpt2plusg.l . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘𝑌))
3 txtopon 21304 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1plusg.g . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
6 tgpcn.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝐺)
7 eqid 2621 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
86, 7tmdtopon 21795 . . . . . . . . . 10 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘(Base‘𝐺)))
95, 8syl 17 . . . . . . . . 9 (𝜑𝐽 ∈ (TopOn‘(Base‘𝐺)))
10 cnmpt2plusg.a . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
11 cnf2 20963 . . . . . . . . 9 (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
124, 9, 10, 11syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
13 eqid 2621 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1413fmpt2 7182 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
1512, 14sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1615r19.21bi 2927 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1716r19.21bi 2927 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
18173impa 1256 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
19 cnmpt2plusg.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
20 cnf2 20963 . . . . . . . . 9 (((𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝐺)) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
214, 9, 19, 20syl3anc 1323 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
22 eqid 2621 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2322fmpt2 7182 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
2421, 23sylibr 224 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2524r19.21bi 2927 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2625r19.21bi 2927 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
27263impa 1256 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
28 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
29 eqid 2621 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
307, 28, 29plusfval 17169 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3118, 27, 30syl2anc 692 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3231mpt2eq3dva 6672 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)))
336, 29tmdcn 21797 . . . 4 (𝐺 ∈ TopMnd → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
345, 33syl 17 . . 3 (𝜑 → (+𝑓𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
351, 2, 10, 19, 34cnmpt22f 21388 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
3632, 35eqeltrrd 2699 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  Basecbs 15781  +gcplusg 15862  TopOpenctopn 16003  +𝑓cplusf 17160  TopOnctopon 20618   Cn ccn 20938   ×t ctx 21273  TopMndctmd 21784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804  df-topgen 16025  df-plusf 17162  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-tx 21275  df-tmd 21786
This theorem is referenced by:  tgpsubcn  21804  oppgtmd  21811  prdstmdd  21837  cnmpt2mulr  21896
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