mndpluscn - Mathbox for Thierry Arnoux

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Theorem mndpluscn 32901

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 6717	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 7539	. . . . 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 22417	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 22417	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 23267	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 7282	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (^◡𝐹‘𝑎) ∈ 𝐵)
14	12, 13	mpan 688	. . . . . . 7 ⊢ (𝑎 ∈ 𝐶 → (^◡𝐹‘𝑎) ∈ 𝐵)
15		f1ocnvdm 7282	. . . . . . . 8 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (^◡𝐹‘𝑏) ∈ 𝐵)
16	12, 15	mpan 688	. . . . . . 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 3197	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		fvoveq1 7431	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((^◡𝐹‘𝑎) + 𝑦)))
21		fveq2 6891	. . . . . . . . 9 ⊢ (𝑥 = (^◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(^◡𝐹‘𝑎)))
22	21	oveq1d 7423	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
23	20, 22	eqeq12d 2748	. . . . . . 7 ⊢ (𝑥 = (^◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((^◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
24		oveq2 7416	. . . . . . . . 9 ⊢ (𝑦 = (^◡𝐹‘𝑏) → ((^◡𝐹‘𝑎) + 𝑦) = ((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))
25	24	fveq2d 6895	. . . . . . . 8 ⊢ (𝑦 = (^◡𝐹‘𝑏) → (𝐹‘((^◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
26		fveq2 6891	. . . . . . . . 9 ⊢ (𝑦 = (^◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑏)))
27	26	oveq2d 7424	. . . . . . . 8 ⊢ (𝑦 = (^◡𝐹‘𝑏) → ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘(^◡𝐹‘𝑏))))
28	25, 27	eqeq12d 2748	. . . . . . 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 3623	. . . . . 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 7274	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑎 ∈ 𝐶) → (𝐹‘(^◡𝐹‘𝑎)) = 𝑎)
32	12, 31	mpan 688	. . . . . 6 ⊢ (𝑎 ∈ 𝐶 → (𝐹‘(^◡𝐹‘𝑎)) = 𝑎)
33		f1ocnvfv2 7274	. . . . . . 7 ⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘(^◡𝐹‘𝑏)) = 𝑏)
34	12, 33	mpan 688	. . . . . 6 ⊢ (𝑏 ∈ 𝐶 → (𝐹‘(^◡𝐹‘𝑏)) = 𝑏)
35	32, 34	oveqan12d 7427	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘(^◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
36	30, 35	eqtr2d 2773	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
37	36	mpoeq3ia 7486	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
38	5, 37	eqtri 2760	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
39	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
40	39, 39	cnmpt1st 23171	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
41		hmeocnvcn 23264	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
42	6, 41	mp1i 13	. . . . . 6 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
43	39, 39, 40, 42	cnmpt21f 23175	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (^◡𝐹‘𝑎)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐽))
44	39, 39	cnmpt2nd 23172	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
45	39, 39, 44, 42	cnmpt21f 23175	. . . . 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 23178	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐽))
49		hmeocn 23263	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
50	6, 49	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
51	39, 39, 48, 50	cnmpt21f 23175	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
52	51	mptru 1548	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
53	38, 52	eqeltri 2829	1 ⊢ ∗ ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ∀wral 3061 × cxp 5674 ^◡ccnv 5675 Fn wfn 6538 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 TopOnctopon 22411 Cn ccn 22727 ×_t ctx 23063 Homeochmeo 23256

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821 df-topgen 17388 df-top 22395 df-topon 22412 df-bases 22448 df-cn 22730 df-tx 23065 df-hmeo 23258

This theorem is referenced by: mhmhmeotmd 32902 xrge0pluscn 32915

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