Theorem mndpluscn 31279

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 6487	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 7261	. . . . 5 ⊢ ( ∗ Fn (𝐶 × 𝐶) ↔ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)))
4	3	biimpi 219	. . . 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 21521	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 21521	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 22369	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 7019	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (◡𝐹‘𝑎) ∈ 𝐵)
14	12, 13	mpan 689	. . . . . . 7 ⊢ (𝑎 ∈ 𝐶 → (◡𝐹‘𝑎) ∈ 𝐵)
15		f1ocnvdm 7019	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (◡𝐹‘𝑏) ∈ 𝐵)
16	12, 15	mpan 689	. . . . . . 7 ⊢ (𝑏 ∈ 𝐶 → (◡𝐹‘𝑏) ∈ 𝐵)
17	14, 16	anim12i 615	. . . . . 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	rgen2 3168	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		fvoveq1 7158	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + 𝑦)))
21		fveq2 6645	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑎)))
22	21	oveq1d 7150	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
23	20, 22	eqeq12d 2814	. . . . . . 7 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
24		oveq2 7143	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((◡𝐹‘𝑎) + 𝑦) = ((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))
25	24	fveq2d 6649	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
26		fveq2 6645	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑏)))
27	26	oveq2d 7151	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
28	25, 27	eqeq12d 2814	. . . . . . 7 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏)))))
29	23, 28	rspc2va 3582	. . . . . 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))))
30	17, 19, 29	sylancl 589	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
31		f1ocnvfv2 7012	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
32	12, 31	mpan 689	. . . . . 6 ⊢ (𝑎 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
33		f1ocnvfv2 7012	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	12, 33	mpan 689	. . . . . 6 ⊢ (𝑏 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	32, 34	oveqan12d 7154	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
36	30, 35	eqtr2d 2834	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
37	36	mpoeq3ia 7211	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
38	5, 37	eqtri 2821	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
39	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
40	39, 39	cnmpt1st 22273	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
41		hmeocnvcn 22366	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
42	6, 41	mp1i 13	. . . . . 6 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn 𝐽))
43	39, 39, 40, 42	cnmpt21f 22277	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
44	39, 39	cnmpt2nd 22274	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
45	39, 39, 44, 42	cnmpt21f 22277	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑏)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
46		mndpluscn.o	. . . . . 6 ⊢ + ∈ ((𝐽 ×t 𝐽) Cn 𝐽)
47	46	a1i 11	. . . . 5 ⊢ (⊤ → + ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
48	39, 39, 43, 45, 47	cnmpt22f 22280	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
49		hmeocn 22365	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
50	6, 49	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
51	39, 39, 48, 50	cnmpt21f 22277	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
52	51	mptru 1545	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
53	38, 52	eqeltri 2886	1 ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  ∀wral 3106   × cxp 5517  ^◡ccnv 5518   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  TopOnctopon 21515   Cn ccn 21829   ×_t ctx 22165  Homeochmeo 22358

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-tx 22167  df-hmeo 22360

This theorem is referenced by:  mhmhmeotmd  31280  xrge0pluscn  31293