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Theorem cnmpt2plusg 23966
Description: Continuity of the group sum; analogue of cnmpt22f 23553 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 23469 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 583 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1plusg.g . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
6 tgpcn.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝐺)
7 eqid 2727 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
86, 7tmdtopon 23959 . . . . . . . . . 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 23127 . . . . . . . . 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 1369 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
13 eqid 2727 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1413fmpo 8064 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
1512, 14sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1615r19.21bi 3243 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1716r19.21bi 3243 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
18173impa 1108 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
19 cnmpt2plusg.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
20 cnf2 23127 . . . . . . . . 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 1369 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
22 eqid 2727 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2322fmpo 8064 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
2421, 23sylibr 233 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2524r19.21bi 3243 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2625r19.21bi 3243 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
27263impa 1108 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
28 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
29 eqid 2727 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
307, 28, 29plusfval 18592 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3118, 27, 30syl2anc 583 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3231mpoeq3dva 7491 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)))
336, 29tmdcn 23961 . . . 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 23553 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
3632, 35eqeltrrd 2829 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056   × cxp 5670  wf 6538  cfv 6542  (class class class)co 7414  cmpo 7416  Basecbs 17165  +gcplusg 17218  TopOpenctopn 17388  +𝑓cplusf 18582  TopOnctopon 22786   Cn ccn 23102   ×t ctx 23438  TopMndctmd 23948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-map 8836  df-topgen 17410  df-plusf 18584  df-top 22770  df-topon 22787  df-topsp 22809  df-bases 22823  df-cn 23105  df-tx 23440  df-tmd 23950
This theorem is referenced by:  tgpsubcn  23968  oppgtmd  23975  prdstmdd  24002  cnmpt2mulr  24061
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