Theorem mhmhmeotmd 34111

Description: Deduce a Topological Monoid using mapping that is both a homeomorphism and a monoid homomorphism. (Contributed by Thierry Arnoux, 21-Jun-2017.)

Hypotheses

Ref	Expression
mhmhmeotmd.m	⊢ 𝐹 ∈ (𝑆 MndHom 𝑇)
mhmhmeotmd.h	⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))
mhmhmeotmd.t	⊢ 𝑆 ∈ TopMnd
mhmhmeotmd.s	⊢ 𝑇 ∈ TopSp

Assertion

Ref	Expression
mhmhmeotmd	⊢ 𝑇 ∈ TopMnd

Proof of Theorem mhmhmeotmd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmhmeotmd.m	. . 3 ⊢ 𝐹 ∈ (𝑆 MndHom 𝑇)
2		mhmrcl2 18747	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑇 ∈ Mnd)
3	1, 2	ax-mp 5	. 2 ⊢ 𝑇 ∈ Mnd
4		mhmhmeotmd.s	. 2 ⊢ 𝑇 ∈ TopSp
5		mhmhmeotmd.h	. . 3 ⊢ 𝐹 ∈ ((TopOpen‘𝑆)Homeo(TopOpen‘𝑇))
6		mhmrcl1 18746	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
7	1, 6	ax-mp 5	. . . 4 ⊢ 𝑆 ∈ Mnd
8		eqid 2739	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2739	. . . . 5 ⊢ (+𝑓‘𝑆) = (+𝑓‘𝑆)
10	8, 9	mndplusf 18711	. . . 4 ⊢ (𝑆 ∈ Mnd → (+𝑓‘𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆))
11	7, 10	ax-mp 5	. . 3 ⊢ (+𝑓‘𝑆):((Base‘𝑆) × (Base‘𝑆))⟶(Base‘𝑆)
12		eqid 2739	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
13		eqid 2739	. . . . 5 ⊢ (+𝑓‘𝑇) = (+𝑓‘𝑇)
14	12, 13	mndplusf 18711	. . . 4 ⊢ (𝑇 ∈ Mnd → (+𝑓‘𝑇):((Base‘𝑇) × (Base‘𝑇))⟶(Base‘𝑇))
15	3, 14	ax-mp 5	. . 3 ⊢ (+𝑓‘𝑇):((Base‘𝑇) × (Base‘𝑇))⟶(Base‘𝑇)
16		mhmhmeotmd.t	. . . 4 ⊢ 𝑆 ∈ TopMnd
17		eqid 2739	. . . . 5 ⊢ (TopOpen‘𝑆) = (TopOpen‘𝑆)
18	17, 8	tmdtopon 24064	. . . 4 ⊢ (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
19	16, 18	ax-mp 5	. . 3 ⊢ (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20		eqid 2739	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
21	12, 20	istps 22917	. . . 4 ⊢ (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
22	4, 21	mpbi 231	. . 3 ⊢ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23		eqid 2739	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2739	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	8, 23, 24	mhmlin 18752	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
26	1, 25	mp3an1 1456	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27	8, 23, 9	plusfval 18606	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓‘𝑆)𝑦) = (𝑥(+g‘𝑆)𝑦))
28	27	fveq2d 6831	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
29	8, 12	mhmf 18748	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
30	1, 29	ax-mp 5	. . . . . 6 ⊢ 𝐹:(Base‘𝑆)⟶(Base‘𝑇)
31	30	ffvelcdmi 7024	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑆) → (𝐹‘𝑥) ∈ (Base‘𝑇))
32	30	ffvelcdmi 7024	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑆) → (𝐹‘𝑦) ∈ (Base‘𝑇))
33	12, 24, 13	plusfval 18606	. . . . 5 ⊢ (((𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
34	31, 32, 33	syl2an 602	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	26, 28, 34	3eqtr4d 2784	. . 3 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)))
36	17, 9	tmdcn 24066	. . . 4 ⊢ (𝑆 ∈ TopMnd → (+𝑓‘𝑆) ∈ (((TopOpen‘𝑆) ×t (TopOpen‘𝑆)) Cn (TopOpen‘𝑆)))
37	16, 36	ax-mp 5	. . 3 ⊢ (+𝑓‘𝑆) ∈ (((TopOpen‘𝑆) ×t (TopOpen‘𝑆)) Cn (TopOpen‘𝑆))
38	5, 11, 15, 19, 22, 35, 37	mndpluscn 34110	. 2 ⊢ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
39	13, 20	istmd 24057	. 2 ⊢ (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
40	3, 4, 38, 39	mpbir3an 1348	1 ⊢ 𝑇 ∈ TopMnd

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119   × cxp 5616  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  +_gcplusg 17211  TopOpenctopn 17375  +_𝑓cplusf 18596  Mndcmnd 18693   MndHom cmhm 18740  TopOnctopon 22893  TopSpctps 22915   Cn ccn 23207   ×_t ctx 23543  Homeochmeo 23736  TopMndctmd 24053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-topgen 17397  df-plusf 18598  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-top 22877  df-topon 22894  df-topsp 22916  df-bases 22929  df-cn 23210  df-tx 23545  df-hmeo 23738  df-tmd 24055

This theorem is referenced by:  xrge0tmd  34129