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Theorem mndpluscn 31226
Description: A mapping that is both a homeomorphism and a monoid homomorphism preserves the "continuousness" of the operation. (Contributed by Thierry Arnoux, 25-Mar-2017.)
Hypotheses
Ref Expression
mndpluscn.f 𝐹 ∈ (𝐽Homeo𝐾)
mndpluscn.p + :(𝐵 × 𝐵)⟶𝐵
mndpluscn.t :(𝐶 × 𝐶)⟶𝐶
mndpluscn.j 𝐽 ∈ (TopOn‘𝐵)
mndpluscn.k 𝐾 ∈ (TopOn‘𝐶)
mndpluscn.h ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
mndpluscn.o + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
Assertion
Ref Expression
mndpluscn ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Distinct variable groups:   𝑦, ,𝑥   𝑦, +   𝑦,𝐹   𝑥, +   𝑥,𝐵,𝑦   𝑥,𝐹
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem mndpluscn
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndpluscn.t . . . 4 :(𝐶 × 𝐶)⟶𝐶
2 ffn 6503 . . . 4 ( :(𝐶 × 𝐶)⟶𝐶 Fn (𝐶 × 𝐶))
3 fnov 7275 . . . . 5 ( Fn (𝐶 × 𝐶) ↔ = (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)))
43biimpi 219 . . . 4 ( Fn (𝐶 × 𝐶) → = (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)))
51, 2, 4mp2b 10 . . 3 = (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏))
6 mndpluscn.f . . . . . . . . 9 𝐹 ∈ (𝐽Homeo𝐾)
7 mndpluscn.j . . . . . . . . . . 11 𝐽 ∈ (TopOn‘𝐵)
87toponunii 21524 . . . . . . . . . 10 𝐵 = 𝐽
9 mndpluscn.k . . . . . . . . . . 11 𝐾 ∈ (TopOn‘𝐶)
109toponunii 21524 . . . . . . . . . 10 𝐶 = 𝐾
118, 10hmeof1o 22372 . . . . . . . . 9 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵1-1-onto𝐶)
126, 11ax-mp 5 . . . . . . . 8 𝐹:𝐵1-1-onto𝐶
13 f1ocnvdm 7033 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹𝑎) ∈ 𝐵)
1412, 13mpan 689 . . . . . . 7 (𝑎𝐶 → (𝐹𝑎) ∈ 𝐵)
15 f1ocnvdm 7033 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹𝑏) ∈ 𝐵)
1612, 15mpan 689 . . . . . . 7 (𝑏𝐶 → (𝐹𝑏) ∈ 𝐵)
1714, 16anim12i 615 . . . . . 6 ((𝑎𝐶𝑏𝐶) → ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵))
18 mndpluscn.h . . . . . . 7 ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918rgen2 3198 . . . . . 6 𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))
20 fvoveq1 7172 . . . . . . . 8 (𝑥 = (𝐹𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((𝐹𝑎) + 𝑦)))
21 fveq2 6661 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝐹𝑥) = (𝐹‘(𝐹𝑎)))
2221oveq1d 7164 . . . . . . . 8 (𝑥 = (𝐹𝑎) → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)))
2320, 22eqeq12d 2840 . . . . . . 7 (𝑥 = (𝐹𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦))))
24 oveq2 7157 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → ((𝐹𝑎) + 𝑦) = ((𝐹𝑎) + (𝐹𝑏)))
2524fveq2d 6665 . . . . . . . 8 (𝑦 = (𝐹𝑏) → (𝐹‘((𝐹𝑎) + 𝑦)) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
26 fveq2 6661 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → (𝐹𝑦) = (𝐹‘(𝐹𝑏)))
2726oveq2d 7165 . . . . . . . 8 (𝑦 = (𝐹𝑏) → ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
2825, 27eqeq12d 2840 . . . . . . 7 (𝑦 = (𝐹𝑏) → ((𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏)))))
2923, 28rspc2va 3620 . . . . . 6 ((((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
3017, 19, 29sylancl 589 . . . . 5 ((𝑎𝐶𝑏𝐶) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
31 f1ocnvfv2 7026 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹‘(𝐹𝑎)) = 𝑎)
3212, 31mpan 689 . . . . . 6 (𝑎𝐶 → (𝐹‘(𝐹𝑎)) = 𝑎)
33 f1ocnvfv2 7026 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹‘(𝐹𝑏)) = 𝑏)
3412, 33mpan 689 . . . . . 6 (𝑏𝐶 → (𝐹‘(𝐹𝑏)) = 𝑏)
3532, 34oveqan12d 7168 . . . . 5 ((𝑎𝐶𝑏𝐶) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
3630, 35eqtr2d 2860 . . . 4 ((𝑎𝐶𝑏𝐶) → (𝑎 𝑏) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
3736mpoeq3ia 7225 . . 3 (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)) = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
385, 37eqtri 2847 . 2 = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
399a1i 11 . . . 4 (⊤ → 𝐾 ∈ (TopOn‘𝐶))
4039, 39cnmpt1st 22276 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
41 hmeocnvcn 22369 . . . . . . 7 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
426, 41mp1i 13 . . . . . 6 (⊤ → 𝐹 ∈ (𝐾 Cn 𝐽))
4339, 39, 40, 42cnmpt21f 22280 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
4439, 39cnmpt2nd 22277 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
4539, 39, 44, 42cnmpt21f 22280 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑏)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
46 mndpluscn.o . . . . . 6 + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
4746a1i 11 . . . . 5 (⊤ → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4839, 39, 43, 45, 47cnmpt22f 22283 . . . 4 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ ((𝐹𝑎) + (𝐹𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
49 hmeocn 22368 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
506, 49mp1i 13 . . . 4 (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
5139, 39, 48, 50cnmpt21f 22280 . . 3 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
5251mptru 1545 . 2 (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
5338, 52eqeltri 2912 1 ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wtru 1539  wcel 2115  wral 3133   × cxp 5540  ccnv 5541   Fn wfn 6338  wf 6339  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  cmpo 7151  TopOnctopon 21518   Cn ccn 21832   ×t ctx 22168  Homeochmeo 22361
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cn 21835  df-tx 22170  df-hmeo 22363
This theorem is referenced by:  mhmhmeotmd  31227  xrge0pluscn  31240
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