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Theorem cnmpt2plusg 22696
Description: Continuity of the group sum; analogue of cnmpt22f 22283 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 22199 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 587 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1plusg.g . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
6 tgpcn.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝐺)
7 eqid 2801 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
86, 7tmdtopon 22689 . . . . . . . . . 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 21857 . . . . . . . . 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 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
13 eqid 2801 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1413fmpo 7752 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
1512, 14sylibr 237 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1615r19.21bi 3176 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1716r19.21bi 3176 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
18173impa 1107 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
19 cnmpt2plusg.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
20 cnf2 21857 . . . . . . . . 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 1368 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
22 eqid 2801 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2322fmpo 7752 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
2421, 23sylibr 237 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2524r19.21bi 3176 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2625r19.21bi 3176 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
27263impa 1107 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
28 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
29 eqid 2801 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
307, 28, 29plusfval 17854 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3118, 27, 30syl2anc 587 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3231mpoeq3dva 7214 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)))
336, 29tmdcn 22691 . . . 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 22283 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
3632, 35eqeltrrd 2894 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109   × cxp 5521  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  Basecbs 16478  +gcplusg 16560  TopOpenctopn 16690  +𝑓cplusf 17844  TopOnctopon 21518   Cn ccn 21832   ×t ctx 22168  TopMndctmd 22678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16712  df-plusf 17846  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-tx 22170  df-tmd 22680
This theorem is referenced by:  tgpsubcn  22698  oppgtmd  22705  prdstmdd  22732  cnmpt2mulr  22791
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