Theorem mhmhmeotmd 33396

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 18708	. . 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 18707	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
7	1, 6	ax-mp 5	. . . 4 ⊢ 𝑆 ∈ Mnd
8		eqid 2724	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2724	. . . . 5 ⊢ (+𝑓‘𝑆) = (+𝑓‘𝑆)
10	8, 9	mndplusf 18675	. . . 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 2724	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
13		eqid 2724	. . . . 5 ⊢ (+𝑓‘𝑇) = (+𝑓‘𝑇)
14	12, 13	mndplusf 18675	. . . 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 2724	. . . . 5 ⊢ (TopOpen‘𝑆) = (TopOpen‘𝑆)
18	17, 8	tmdtopon 23907	. . . 4 ⊢ (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
19	16, 18	ax-mp 5	. . 3 ⊢ (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20		eqid 2724	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
21	12, 20	istps 22758	. . . 4 ⊢ (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
22	4, 21	mpbi 229	. . 3 ⊢ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23		eqid 2724	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2724	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	8, 23, 24	mhmlin 18713	. . . . 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 1444	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27	8, 23, 9	plusfval 18570	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓‘𝑆)𝑦) = (𝑥(+g‘𝑆)𝑦))
28	27	fveq2d 6885	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
29	8, 12	mhmf 18709	. . . . . . 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 7075	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑆) → (𝐹‘𝑥) ∈ (Base‘𝑇))
32	30	ffvelcdmi 7075	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑆) → (𝐹‘𝑦) ∈ (Base‘𝑇))
33	12, 24, 13	plusfval 18570	. . . . 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 595	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	26, 28, 34	3eqtr4d 2774	. . 3 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)))
36	17, 9	tmdcn 23909	. . . 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 33395	. 2 ⊢ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
39	13, 20	istmd 23900	. 2 ⊢ (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
40	3, 4, 38, 39	mpbir3an 1338	1 ⊢ 𝑇 ∈ TopMnd

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098   × cxp 5664  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  Basecbs 17143  +_gcplusg 17196  TopOpenctopn 17366  +_𝑓cplusf 18560  Mndcmnd 18657   MndHom cmhm 18701  TopOnctopon 22734  TopSpctps 22756   Cn ccn 23050   ×_t ctx 23386  Homeochmeo 23579  TopMndctmd 23896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-map 8818  df-topgen 17388  df-plusf 18562  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18703  df-top 22718  df-topon 22735  df-topsp 22757  df-bases 22771  df-cn 23053  df-tx 23388  df-hmeo 23581  df-tmd 23898

This theorem is referenced by:  xrge0tmd  33414