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Theorem mndpluscn 29796
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 6012 . . . 4 ( :(𝐶 × 𝐶)⟶𝐶 Fn (𝐶 × 𝐶))
3 fnov 6733 . . . . 5 ( Fn (𝐶 × 𝐶) ↔ = (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)))
43biimpi 206 . . . 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 20661 . . . . . . . . . 10 𝐵 = 𝐽
9 mndpluscn.k . . . . . . . . . . 11 𝐾 ∈ (TopOn‘𝐶)
109toponunii 20661 . . . . . . . . . 10 𝐶 = 𝐾
118, 10hmeof1o 21507 . . . . . . . . 9 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵1-1-onto𝐶)
126, 11ax-mp 5 . . . . . . . 8 𝐹:𝐵1-1-onto𝐶
13 f1ocnvdm 6505 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹𝑎) ∈ 𝐵)
1412, 13mpan 705 . . . . . . 7 (𝑎𝐶 → (𝐹𝑎) ∈ 𝐵)
15 f1ocnvdm 6505 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹𝑏) ∈ 𝐵)
1612, 15mpan 705 . . . . . . 7 (𝑏𝐶 → (𝐹𝑏) ∈ 𝐵)
1714, 16anim12i 589 . . . . . 6 ((𝑎𝐶𝑏𝐶) → ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵))
18 mndpluscn.h . . . . . . 7 ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918rgen2a 2973 . . . . . 6 𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))
20 oveq1 6622 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝑥 + 𝑦) = ((𝐹𝑎) + 𝑦))
2120fveq2d 6162 . . . . . . . 8 (𝑥 = (𝐹𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((𝐹𝑎) + 𝑦)))
22 fveq2 6158 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝐹𝑥) = (𝐹‘(𝐹𝑎)))
2322oveq1d 6630 . . . . . . . 8 (𝑥 = (𝐹𝑎) → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)))
2421, 23eqeq12d 2636 . . . . . . 7 (𝑥 = (𝐹𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦))))
25 oveq2 6623 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → ((𝐹𝑎) + 𝑦) = ((𝐹𝑎) + (𝐹𝑏)))
2625fveq2d 6162 . . . . . . . 8 (𝑦 = (𝐹𝑏) → (𝐹‘((𝐹𝑎) + 𝑦)) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
27 fveq2 6158 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → (𝐹𝑦) = (𝐹‘(𝐹𝑏)))
2827oveq2d 6631 . . . . . . . 8 (𝑦 = (𝐹𝑏) → ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
2926, 28eqeq12d 2636 . . . . . . 7 (𝑦 = (𝐹𝑏) → ((𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏)))))
3024, 29rspc2va 3312 . . . . . 6 ((((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
3117, 19, 30sylancl 693 . . . . 5 ((𝑎𝐶𝑏𝐶) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
32 f1ocnvfv2 6498 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹‘(𝐹𝑎)) = 𝑎)
3312, 32mpan 705 . . . . . 6 (𝑎𝐶 → (𝐹‘(𝐹𝑎)) = 𝑎)
34 f1ocnvfv2 6498 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹‘(𝐹𝑏)) = 𝑏)
3512, 34mpan 705 . . . . . 6 (𝑏𝐶 → (𝐹‘(𝐹𝑏)) = 𝑏)
3633, 35oveqan12d 6634 . . . . 5 ((𝑎𝐶𝑏𝐶) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
3731, 36eqtr2d 2656 . . . 4 ((𝑎𝐶𝑏𝐶) → (𝑎 𝑏) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
3837mpt2eq3ia 6685 . . 3 (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)) = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
395, 38eqtri 2643 . 2 = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
409a1i 11 . . . 4 (⊤ → 𝐾 ∈ (TopOn‘𝐶))
4140, 40cnmpt1st 21411 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
42 hmeocnvcn 21504 . . . . . . 7 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
436, 42mp1i 13 . . . . . 6 (⊤ → 𝐹 ∈ (𝐾 Cn 𝐽))
4440, 40, 41, 43cnmpt21f 21415 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
4540, 40cnmpt2nd 21412 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
4640, 40, 45, 43cnmpt21f 21415 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑏)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
47 mndpluscn.o . . . . . 6 + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
4847a1i 11 . . . . 5 (⊤ → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
4940, 40, 44, 46, 48cnmpt22f 21418 . . . 4 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ ((𝐹𝑎) + (𝐹𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
50 hmeocn 21503 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
516, 50mp1i 13 . . . 4 (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
5240, 40, 49, 51cnmpt21f 21415 . . 3 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
5352trud 1490 . 2 (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
5439, 53eqeltri 2694 1 ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wtru 1481  wcel 1987  wral 2908   × cxp 5082  ccnv 5083   Fn wfn 5852  wf 5853  1-1-ontowf1o 5856  cfv 5857  (class class class)co 6615  cmpt2 6617  TopOnctopon 20655   Cn ccn 20968   ×t ctx 21303  Homeochmeo 21496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-map 7819  df-topgen 16044  df-top 20639  df-topon 20656  df-bases 20690  df-cn 20971  df-tx 21305  df-hmeo 21498
This theorem is referenced by:  mhmhmeotmd  29797  xrge0pluscn  29810
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