Theorem mndpluscn 31876

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 6600	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 7405	. . . . 5 ⊢ ( ∗ Fn (𝐶 × 𝐶) ↔ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)))
4	3	biimpi 215	. . . 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 22065	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 22065	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 22915	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 7157	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (◡𝐹‘𝑎) ∈ 𝐵)
14	12, 13	mpan 687	. . . . . . 7 ⊢ (𝑎 ∈ 𝐶 → (◡𝐹‘𝑎) ∈ 𝐵)
15		f1ocnvdm 7157	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (◡𝐹‘𝑏) ∈ 𝐵)
16	12, 15	mpan 687	. . . . . . 7 ⊢ (𝑏 ∈ 𝐶 → (◡𝐹‘𝑏) ∈ 𝐵)
17	14, 16	anim12i 613	. . . . . 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 3120	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		fvoveq1 7298	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + 𝑦)))
21		fveq2 6774	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑎)))
22	21	oveq1d 7290	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
23	20, 22	eqeq12d 2754	. . . . . . 7 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
24		oveq2 7283	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((◡𝐹‘𝑎) + 𝑦) = ((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))
25	24	fveq2d 6778	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
26		fveq2 6774	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑏)))
27	26	oveq2d 7291	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
28	25, 27	eqeq12d 2754	. . . . . . 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 3571	. . . . . 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 586	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
31		f1ocnvfv2 7149	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
32	12, 31	mpan 687	. . . . . 6 ⊢ (𝑎 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
33		f1ocnvfv2 7149	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	12, 33	mpan 687	. . . . . 6 ⊢ (𝑏 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	32, 34	oveqan12d 7294	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
36	30, 35	eqtr2d 2779	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
37	36	mpoeq3ia 7353	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
38	5, 37	eqtri 2766	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
39	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
40	39, 39	cnmpt1st 22819	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
41		hmeocnvcn 22912	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
42	6, 41	mp1i 13	. . . . . 6 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn 𝐽))
43	39, 39, 40, 42	cnmpt21f 22823	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
44	39, 39	cnmpt2nd 22820	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
45	39, 39, 44, 42	cnmpt21f 22823	. . . . 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 22826	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
49		hmeocn 22911	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
50	6, 49	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
51	39, 39, 48, 50	cnmpt21f 22823	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
52	51	mptru 1546	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
53	38, 52	eqeltri 2835	1 ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ∀wral 3064   × cxp 5587  ^◡ccnv 5588   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  TopOnctopon 22059   Cn ccn 22375   ×_t ctx 22711  Homeochmeo 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-tx 22713  df-hmeo 22906

This theorem is referenced by:  mhmhmeotmd  31877  xrge0pluscn  31890