mndpluscn - Mathbox for Thierry Arnoux

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Theorem mndpluscn 32906

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 6718	. . . 4 ⊢ ( ∗ :(𝐶 × 𝐶)⟶𝐶 → ∗ Fn (𝐶 × 𝐶))
3		fnov 7540	. . . . 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 22418	. . . . . . . . . 10 ⊢ 𝐵 = ∪ 𝐽
9		mndpluscn.k	. . . . . . . . . . 11 ⊢ 𝐾 ∈ (TopOn‘𝐶)
10	9	toponunii 22418	. . . . . . . . . 10 ⊢ 𝐶 = ∪ 𝐾
11	8, 10	hmeof1o 23268	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝐵–1-1-onto→𝐶)
12	6, 11	ax-mp 5	. . . . . . . 8 ⊢ 𝐹:𝐵–1-1-onto→𝐶
13		f1ocnvdm 7283	. . . . . . . 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 7283	. . . . . . . 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 614	. . . . . 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 3198	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦))
20		fvoveq1 7432	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹‘𝑎) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘((^◡𝐹‘𝑎) + 𝑦)))
21		fveq2 6892	. . . . . . . . 9 ⊢ (𝑥 = (^◡𝐹‘𝑎) → (𝐹‘𝑥) = (𝐹‘(^◡𝐹‘𝑎)))
22	21	oveq1d 7424	. . . . . . . 8 ⊢ (𝑥 = (^◡𝐹‘𝑎) → ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)))
23	20, 22	eqeq12d 2749	. . . . . . 7 ⊢ (𝑥 = (^◡𝐹‘𝑎) → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ∗ (𝐹‘𝑦)) ↔ (𝐹‘((^◡𝐹‘𝑎) + 𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦))))
24		oveq2 7417	. . . . . . . . 9 ⊢ (𝑦 = (^◡𝐹‘𝑏) → ((^◡𝐹‘𝑎) + 𝑦) = ((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))
25	24	fveq2d 6896	. . . . . . . 8 ⊢ (𝑦 = (^◡𝐹‘𝑏) → (𝐹‘((^◡𝐹‘𝑎) + 𝑦)) = (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
26		fveq2 6892	. . . . . . . . 9 ⊢ (𝑦 = (^◡𝐹‘𝑏) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑏)))
27	26	oveq2d 7425	. . . . . . . 8 ⊢ (𝑦 = (^◡𝐹‘𝑏) → ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘𝑦)) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘(^◡𝐹‘𝑏))))
28	25, 27	eqeq12d 2749	. . . . . . 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 3624	. . . . . 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 587	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))) = ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘(^◡𝐹‘𝑏))))
31		f1ocnvfv2 7275	. . . . . . 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 7275	. . . . . . 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 7428	. . . . 5 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → ((𝐹‘(^◡𝐹‘𝑎)) ∗ (𝐹‘(^◡𝐹‘𝑏))) = (𝑎 ∗ 𝑏))
36	30, 35	eqtr2d 2774	. . . 4 ⊢ ((𝑎 ∈ 𝐶 ∧ 𝑏 ∈ 𝐶) → (𝑎 ∗ 𝑏) = (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
37	36	mpoeq3ia 7487	. . 3 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝑎 ∗ 𝑏)) = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
38	5, 37	eqtri 2761	. 2 ⊢ ∗ = (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))))
39	9	a1i 11	. . . 4 ⊢ (⊤ → 𝐾 ∈ (TopOn‘𝐶))
40	39, 39	cnmpt1st 23172	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑎) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
41		hmeocnvcn 23265	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
42	6, 41	mp1i 13	. . . . . 6 ⊢ (⊤ → ^◡𝐹 ∈ (𝐾 Cn 𝐽))
43	39, 39, 40, 42	cnmpt21f 23176	. . . . 5 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (^◡𝐹‘𝑎)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐽))
44	39, 39	cnmpt2nd 23173	. . . . . 6 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ 𝑏) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
45	39, 39, 44, 42	cnmpt21f 23176	. . . . 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 23179	. . . 4 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ ((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐽))
49		hmeocn 23264	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
50	6, 49	mp1i 13	. . . 4 ⊢ (⊤ → 𝐹 ∈ (𝐽 Cn 𝐾))
51	39, 39, 48, 50	cnmpt21f 23176	. . 3 ⊢ (⊤ → (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
52	51	mptru 1549	. 2 ⊢ (𝑎 ∈ 𝐶, 𝑏 ∈ 𝐶 ↦ (𝐹‘((^◡𝐹‘𝑎) + (^◡𝐹‘𝑏)))) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
53	38, 52	eqeltri 2830	1 ⊢ ∗ ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ∀wral 3062 × cxp 5675 ^◡ccnv 5676 Fn wfn 6539 ⟶wf 6540 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 TopOnctopon 22412 Cn ccn 22728 ×_t ctx 23064 Homeochmeo 23257

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-topgen 17389 df-top 22396 df-topon 22413 df-bases 22449 df-cn 22731 df-tx 23066 df-hmeo 23259

This theorem is referenced by: mhmhmeotmd 32907 xrge0pluscn 32920

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