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Theorem mndpluscn 31068
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 6507 . . . 4 ( :(𝐶 × 𝐶)⟶𝐶 Fn (𝐶 × 𝐶))
3 fnov 7271 . . . . 5 ( Fn (𝐶 × 𝐶) ↔ = (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)))
43biimpi 217 . . . 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 21452 . . . . . . . . . 10 𝐵 = 𝐽
9 mndpluscn.k . . . . . . . . . . 11 𝐾 ∈ (TopOn‘𝐶)
109toponunii 21452 . . . . . . . . . 10 𝐶 = 𝐾
118, 10hmeof1o 22300 . . . . . . . . 9 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵1-1-onto𝐶)
126, 11ax-mp 5 . . . . . . . 8 𝐹:𝐵1-1-onto𝐶
13 f1ocnvdm 7032 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹𝑎) ∈ 𝐵)
1412, 13mpan 686 . . . . . . 7 (𝑎𝐶 → (𝐹𝑎) ∈ 𝐵)
15 f1ocnvdm 7032 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹𝑏) ∈ 𝐵)
1612, 15mpan 686 . . . . . . 7 (𝑏𝐶 → (𝐹𝑏) ∈ 𝐵)
1714, 16anim12i 612 . . . . . 6 ((𝑎𝐶𝑏𝐶) → ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵))
18 mndpluscn.h . . . . . . 7 ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918rgen2 3200 . . . . . 6 𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))
20 fvoveq1 7168 . . . . . . . 8 (𝑥 = (𝐹𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((𝐹𝑎) + 𝑦)))
21 fveq2 6663 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝐹𝑥) = (𝐹‘(𝐹𝑎)))
2221oveq1d 7160 . . . . . . . 8 (𝑥 = (𝐹𝑎) → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)))
2320, 22eqeq12d 2834 . . . . . . 7 (𝑥 = (𝐹𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦))))
24 oveq2 7153 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → ((𝐹𝑎) + 𝑦) = ((𝐹𝑎) + (𝐹𝑏)))
2524fveq2d 6667 . . . . . . . 8 (𝑦 = (𝐹𝑏) → (𝐹‘((𝐹𝑎) + 𝑦)) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
26 fveq2 6663 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → (𝐹𝑦) = (𝐹‘(𝐹𝑏)))
2726oveq2d 7161 . . . . . . . 8 (𝑦 = (𝐹𝑏) → ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
2825, 27eqeq12d 2834 . . . . . . 7 (𝑦 = (𝐹𝑏) → ((𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏)))))
2923, 28rspc2va 3631 . . . . . 6 ((((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
3017, 19, 29sylancl 586 . . . . 5 ((𝑎𝐶𝑏𝐶) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
31 f1ocnvfv2 7025 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹‘(𝐹𝑎)) = 𝑎)
3212, 31mpan 686 . . . . . 6 (𝑎𝐶 → (𝐹‘(𝐹𝑎)) = 𝑎)
33 f1ocnvfv2 7025 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹‘(𝐹𝑏)) = 𝑏)
3412, 33mpan 686 . . . . . 6 (𝑏𝐶 → (𝐹‘(𝐹𝑏)) = 𝑏)
3532, 34oveqan12d 7164 . . . . 5 ((𝑎𝐶𝑏𝐶) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
3630, 35eqtr2d 2854 . . . 4 ((𝑎𝐶𝑏𝐶) → (𝑎 𝑏) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
3736mpoeq3ia 7221 . . 3 (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)) = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
385, 37eqtri 2841 . 2 = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
399a1i 11 . . . 4 (⊤ → 𝐾 ∈ (TopOn‘𝐶))
4039, 39cnmpt1st 22204 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
41 hmeocnvcn 22297 . . . . . . 7 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
426, 41mp1i 13 . . . . . 6 (⊤ → 𝐹 ∈ (𝐾 Cn 𝐽))
4339, 39, 40, 42cnmpt21f 22208 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
4439, 39cnmpt2nd 22205 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
4539, 39, 44, 42cnmpt21f 22208 . . . . 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 22211 . . . 4 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ ((𝐹𝑎) + (𝐹𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
49 hmeocn 22296 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
506, 49mp1i 13 . . . 4 (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
5139, 39, 48, 50cnmpt21f 22208 . . 3 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
5251mptru 1535 . 2 (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
5338, 52eqeltri 2906 1 ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wtru 1529  wcel 2105  wral 3135   × cxp 5546  ccnv 5547   Fn wfn 6343  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  TopOnctopon 21446   Cn ccn 21760   ×t ctx 22096  Homeochmeo 22289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-tx 22098  df-hmeo 22291
This theorem is referenced by:  mhmhmeotmd  31069  xrge0pluscn  31082
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