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Theorem mndpluscn 30303
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 6207 . . . 4 ( :(𝐶 × 𝐶)⟶𝐶 Fn (𝐶 × 𝐶))
3 fnov 6935 . . . . 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 20944 . . . . . . . . . 10 𝐵 = 𝐽
9 mndpluscn.k . . . . . . . . . . 11 𝐾 ∈ (TopOn‘𝐶)
109toponunii 20944 . . . . . . . . . 10 𝐶 = 𝐾
118, 10hmeof1o 21790 . . . . . . . . 9 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵1-1-onto𝐶)
126, 11ax-mp 5 . . . . . . . 8 𝐹:𝐵1-1-onto𝐶
13 f1ocnvdm 6705 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹𝑎) ∈ 𝐵)
1412, 13mpan 708 . . . . . . 7 (𝑎𝐶 → (𝐹𝑎) ∈ 𝐵)
15 f1ocnvdm 6705 . . . . . . . 8 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹𝑏) ∈ 𝐵)
1612, 15mpan 708 . . . . . . 7 (𝑏𝐶 → (𝐹𝑏) ∈ 𝐵)
1714, 16anim12i 591 . . . . . 6 ((𝑎𝐶𝑏𝐶) → ((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵))
18 mndpluscn.h . . . . . . 7 ((𝑥𝐵𝑦𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
1918rgen2a 3116 . . . . . 6 𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))
20 oveq1 6822 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝑥 + 𝑦) = ((𝐹𝑎) + 𝑦))
2120fveq2d 6358 . . . . . . . 8 (𝑥 = (𝐹𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((𝐹𝑎) + 𝑦)))
22 fveq2 6354 . . . . . . . . 9 (𝑥 = (𝐹𝑎) → (𝐹𝑥) = (𝐹‘(𝐹𝑎)))
2322oveq1d 6830 . . . . . . . 8 (𝑥 = (𝐹𝑎) → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)))
2421, 23eqeq12d 2776 . . . . . . 7 (𝑥 = (𝐹𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦))))
25 oveq2 6823 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → ((𝐹𝑎) + 𝑦) = ((𝐹𝑎) + (𝐹𝑏)))
2625fveq2d 6358 . . . . . . . 8 (𝑦 = (𝐹𝑏) → (𝐹‘((𝐹𝑎) + 𝑦)) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
27 fveq2 6354 . . . . . . . . 9 (𝑦 = (𝐹𝑏) → (𝐹𝑦) = (𝐹‘(𝐹𝑏)))
2827oveq2d 6831 . . . . . . . 8 (𝑦 = (𝐹𝑏) → ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
2926, 28eqeq12d 2776 . . . . . . 7 (𝑦 = (𝐹𝑏) → ((𝐹‘((𝐹𝑎) + 𝑦)) = ((𝐹‘(𝐹𝑎)) (𝐹𝑦)) ↔ (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏)))))
3024, 29rspc2va 3463 . . . . . 6 ((((𝐹𝑎) ∈ 𝐵 ∧ (𝐹𝑏) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
3117, 19, 30sylancl 697 . . . . 5 ((𝑎𝐶𝑏𝐶) → (𝐹‘((𝐹𝑎) + (𝐹𝑏))) = ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))))
32 f1ocnvfv2 6698 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑎𝐶) → (𝐹‘(𝐹𝑎)) = 𝑎)
3312, 32mpan 708 . . . . . 6 (𝑎𝐶 → (𝐹‘(𝐹𝑎)) = 𝑎)
34 f1ocnvfv2 6698 . . . . . . 7 ((𝐹:𝐵1-1-onto𝐶𝑏𝐶) → (𝐹‘(𝐹𝑏)) = 𝑏)
3512, 34mpan 708 . . . . . 6 (𝑏𝐶 → (𝐹‘(𝐹𝑏)) = 𝑏)
3633, 35oveqan12d 6834 . . . . 5 ((𝑎𝐶𝑏𝐶) → ((𝐹‘(𝐹𝑎)) (𝐹‘(𝐹𝑏))) = (𝑎 𝑏))
3731, 36eqtr2d 2796 . . . 4 ((𝑎𝐶𝑏𝐶) → (𝑎 𝑏) = (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
3837mpt2eq3ia 6887 . . 3 (𝑎𝐶, 𝑏𝐶 ↦ (𝑎 𝑏)) = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
395, 38eqtri 2783 . 2 = (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏))))
409a1i 11 . . . 4 (⊤ → 𝐾 ∈ (TopOn‘𝐶))
4140, 40cnmpt1st 21694 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
42 hmeocnvcn 21787 . . . . . . 7 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
436, 42mp1i 13 . . . . . 6 (⊤ → 𝐹 ∈ (𝐾 Cn 𝐽))
4440, 40, 41, 43cnmpt21f 21698 . . . . 5 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
4540, 40cnmpt2nd 21695 . . . . . 6 (⊤ → (𝑎𝐶, 𝑏𝐶𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
4640, 40, 45, 43cnmpt21f 21698 . . . . 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 21701 . . . 4 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ ((𝐹𝑎) + (𝐹𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
50 hmeocn 21786 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
516, 50mp1i 13 . . . 4 (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
5240, 40, 49, 51cnmpt21f 21698 . . 3 (⊤ → (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
5352trud 1642 . 2 (𝑎𝐶, 𝑏𝐶 ↦ (𝐹‘((𝐹𝑎) + (𝐹𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
5439, 53eqeltri 2836 1 ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wtru 1633  wcel 2140  wral 3051   × cxp 5265  ccnv 5266   Fn wfn 6045  wf 6046  1-1-ontowf1o 6049  cfv 6050  (class class class)co 6815  cmpt2 6817  TopOnctopon 20938   Cn ccn 21251   ×t ctx 21586  Homeochmeo 21779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-map 8028  df-topgen 16327  df-top 20922  df-topon 20939  df-bases 20973  df-cn 21254  df-tx 21588  df-hmeo 21781
This theorem is referenced by:  mhmhmeotmd  30304  xrge0pluscn  30317
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