Theorem mndpluscn 34183

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 6685	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 7521	. . . . 5 ⊢ ( ∗ Fn (𝐶 × 𝐶) ↔ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)))
4	3	biimpi 218	. . . 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 22963	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 22963	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 23811	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 7263	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (◡𝐹‘𝑎) ∈ 𝐵)
14	12, 13	mpan 700	. . . . . . 7 ⊢ (𝑎 ∈ 𝐶 → (◡𝐹‘𝑎) ∈ 𝐵)
15		f1ocnvdm 7263	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (◡𝐹‘𝑏) ∈ 𝐵)
16	12, 15	mpan 700	. . . . . . 7 ⊢ (𝑏 ∈ 𝐶 → (◡𝐹‘𝑏) ∈ 𝐵)
17	14, 16	anim12i 622	. . . . . 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 3201	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		fvoveq1 7413	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + 𝑦)))
21		fveq2 6861	. . . . . . . . 9 ⊢ (𝑥 = (◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑎)))
22	21	oveq1d 7405	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
23	20, 22	eqeq12d 2777	. . . . . . 7 ⊢ (𝑥 = (◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
24		oveq2 7398	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((◡𝐹‘𝑎) + 𝑦) = ((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))
25	24	fveq2d 6865	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘((◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
26		fveq2 6861	. . . . . . . . 9 ⊢ (𝑦 = (◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑏)))
27	26	oveq2d 7406	. . . . . . . 8 ⊢ (𝑦 = (◡𝐹‘𝑏) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
28	25, 27	eqeq12d 2777	. . . . . . 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 3592	. . . . . 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 595	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) = ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))))
31		f1ocnvfv2 7255	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
32	12, 31	mpan 700	. . . . . 6 ⊢ (𝑎 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑎)) = 𝑎)
33		f1ocnvfv2 7255	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
34	12, 33	mpan 700	. . . . . 6 ⊢ (𝑏 ∈ 𝐶 → (𝐹‘(◡𝐹‘𝑏)) = 𝑏)
35	32, 34	oveqan12d 7409	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(◡𝐹‘𝑎)) ∗ (𝐹‘(◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
36	30, 35	eqtr2d 2797	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
37	36	mpoeq3ia 7468	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
38	5, 37	eqtri 2784	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏))))
39	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
40	39, 39	cnmpt1st 23715	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
41		hmeocnvcn 23808	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
42	6, 41	mp1i 13	. . . . . 6 ⊢ (⊤ → ◡𝐹 ∈ (𝐾 Cn 𝐽))
43	39, 39, 40, 42	cnmpt21f 23719	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (◡𝐹‘𝑎)) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
44	39, 39	cnmpt2nd 23716	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
45	39, 39, 44, 42	cnmpt21f 23719	. . . . 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 23722	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((◡𝐹‘𝑎) + (◡𝐹‘𝑏))) ∈ ((𝐾 ×t 𝐾) Cn 𝐽))
49		hmeocn 23807	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
50	6, 49	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
51	39, 39, 48, 50	cnmpt21f 23719	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾))
52	51	mptru 1566	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((◡𝐹‘𝑎) + (◡𝐹‘𝑏)))) ∈ ((𝐾 ×t 𝐾) Cn 𝐾)
53	38, 52	eqeltri 2857	1 ⊢ ∗ ∈ ((𝐾 ×t 𝐾) Cn 𝐾)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141  ∀wral 3075   × cxp 5641  ^◡ccnv 5642   Fn wfn 6510  ⟶wf 6511  –1-1-onto→wf1o 6514  ‘cfv 6515  (class class class)co 7390   ∈ cmpo 7392  TopOnctopon 22957   Cn ccn 23271   ×_t ctx 23607  Homeochmeo 23800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7964  df-2nd 7965  df-map 8803  df-topgen 17462  df-top 22941  df-topon 22958  df-bases 22993  df-cn 23274  df-tx 23609  df-hmeo 23802

This theorem is referenced by:  mhmhmeotmd  34184  xrge0pluscn  34197