MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt2plusg Structured version   Visualization version   GIF version

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 + = (+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 21772 . . . . . . . . . 10 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
41, 2, 3syl2anc 579 . . . . . . . . 9 (𝜑 → (𝐾 ×t 𝐿) ∈ (TopOn‘(𝑋 × 𝑌)))
5 cnmpt1plusg.g . . . . . . . . . 10 (𝜑𝐺 ∈ TopMnd)
6 tgpcn.j . . . . . . . . . . 11 𝐽 = (TopOpen‘𝐺)
7 eqid 2825 . . . . . . . . . . 11 (Base‘𝐺) = (Base‘𝐺)
86, 7tmdtopon 22262 . . . . . . . . . 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 21431 . . . . . . . . 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 1494 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
13 eqid 2825 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
1413fmpt2 7505 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶(Base‘𝐺))
1512, 14sylibr 226 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1615r19.21bi 3141 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐴 ∈ (Base‘𝐺))
1716r19.21bi 3141 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
18173impa 1140 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 ∈ (Base‘𝐺))
19 cnmpt2plusg.b . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
20 cnf2 21431 . . . . . . . . 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 1494 . . . . . . . 8 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
22 eqid 2825 . . . . . . . . 9 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
2322fmpt2 7505 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺) ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶(Base‘𝐺))
2421, 23sylibr 226 . . . . . . 7 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2524r19.21bi 3141 . . . . . 6 ((𝜑𝑥𝑋) → ∀𝑦𝑌 𝐵 ∈ (Base‘𝐺))
2625r19.21bi 3141 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
27263impa 1140 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 ∈ (Base‘𝐺))
28 cnmpt1plusg.p . . . . 5 + = (+g𝐺)
29 eqid 2825 . . . . 5 (+𝑓𝐺) = (+𝑓𝐺)
307, 28, 29plusfval 17608 . . . 4 ((𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3118, 27, 30syl2anc 579 . . 3 ((𝜑𝑥𝑋𝑦𝑌) → (𝐴(+𝑓𝐺)𝐵) = (𝐴 + 𝐵))
3231mpt2eq3dva 6984 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) = (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)))
336, 29tmdcn 22264 . . . 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 21856 . 2 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴(+𝑓𝐺)𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
3632, 35eqeltrrd 2907 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 + 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator