Theorem mndpluscn 29300

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	⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))
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.

Step	Hyp	Ref	Expression
1		mndpluscn.t	. . . 4 ⊢ ∗ :(𝐶 × 𝐶)⟶𝐶
2		ffn 5958	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 6666	. . . . 5 ⊢ ( ∗ Fn (𝐶 × 𝐶) ↔ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)))
4	3	biimpi 205	. . . 4 ⊢ ( ∗ Fn (𝐶 × 𝐶) → ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)))
5	1, 2, 4	mp2b 10	. . 3 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏))
6		mndpluscn.f	. . . . . . . . 9 ⊢ 𝐹 ∈ (𝐽Homeo𝐾)
7		mndpluscn.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ (TopOn‘𝐵)
8	7	toponunii 20547	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 20547	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 21377	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 6440	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (◡𝐹‘𝑎) ∈ 𝐵)
14	12, 13	mpan 702	. . . . . . 7 ⊢ (𝑎 ∈ 𝐶 → (◡𝐹‘𝑎) ∈ 𝐵)
15		f1ocnvdm 6440	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (◡𝐹‘𝑏) ∈ 𝐵)
16	12, 15	mpan 702	. . . . . . 7 ⊢ (𝑏 ∈ 𝐶 → (◡𝐹‘𝑏) ∈ 𝐵)
17	14, 16	anim12i 588	. . . . . 6 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((◡𝐹‘𝑎) ∈ 𝐵 ∧ (◡𝐹‘𝑏) ∈ 𝐵))
18		mndpluscn.h	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)))
19	18	rgen2a 2960	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		oveq1 6556	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝑥 + 𝑦) = ((◡𝐹‘𝑎) + 𝑦))
21	20	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + 𝑦)))
22		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑎)))
23	22	oveq1d 6564	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
24	21, 23	eqeq12d 2625	. . . . . . 7 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
25		oveq2 6557	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((◡𝐹‘𝑎) + 𝑦) = ((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))
26	25	fveq2d 6107	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
27		fveq2 6103	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑏)))
28	27	oveq2d 6565	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
29	26, 28	eqeq12d 2625	. . . . . . 7 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏)))))
30	24, 29	rspc2va 3294	. . . . . 6 ⊢ ((((◡𝐹‘𝑎) ∈ 𝐵 ∧ (◡𝐹‘𝑏) ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))) → (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
31	17, 19, 30	sylancl 693	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
32		f1ocnvfv2 6433	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
33	12, 32	mpan 702	. . . . . 6 ⊢ (𝑎 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
34		f1ocnvfv2 6433	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	12, 34	mpan 702	. . . . . 6 ⊢ (𝑏 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
36	33, 35	oveqan12d 6568	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
37	31, 36	eqtr2d 2645	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
38	37	mpt2eq3ia 6618	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
39	5, 38	eqtri 2632	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
40	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
41	40, 40	cnmpt1st 21281	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
42		hmeocnvcn 21374	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
43	6, 42	mp1i 13	. . . . . 6 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn 𝐽))
44	40, 40, 41, 43	cnmpt21f 21285	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
45	40, 40	cnmpt2nd 21282	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
46	40, 40, 45, 43	cnmpt21f 21285	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑏)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
47		mndpluscn.o	. . . . . 6 ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
48	47	a1i 11	. . . . 5 ⊢ (⊤ → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
49	40, 40, 44, 46, 48	cnmpt22f 21288	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
50		hmeocn 21373	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
51	6, 50	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
52	40, 40, 49, 51	cnmpt21f 21285	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
53	52	trud 1484	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
54	39, 53	eqeltri 2684	1 ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  ∀wral 2896   × cxp 5036  ^◡ccnv 5037   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  TopOnctopon 20518   Cn ccn 20838   ×_t ctx 21173  Homeochmeo 21366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-tx 21175  df-hmeo 21368

This theorem is referenced by:  mhmhmeotmd  29301  xrge0pluscn  29314