Theorem mhmhmeotmd 34185

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 18805	. . 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 18804	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
7	1, 6	ax-mp 5	. . . 4 ⊢ 𝑆 ∈ Mnd
8		eqid 2761	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2761	. . . . 5 ⊢ (+𝑓‘𝑆) = (+𝑓‘𝑆)
10	8, 9	mndplusf 18769	. . . 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 2761	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
13		eqid 2761	. . . . 5 ⊢ (+𝑓‘𝑇) = (+𝑓‘𝑇)
14	12, 13	mndplusf 18769	. . . 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 2761	. . . . 5 ⊢ (TopOpen‘𝑆) = (TopOpen‘𝑆)
18	17, 8	tmdtopon 24121	. . . 4 ⊢ (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
19	16, 18	ax-mp 5	. . 3 ⊢ (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20		eqid 2761	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
21	12, 20	istps 22974	. . . 4 ⊢ (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
22	4, 21	mpbi 232	. . 3 ⊢ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23		eqid 2761	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2761	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	8, 23, 24	mhmlin 18810	. . . . 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 1468	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27	8, 23, 9	plusfval 18664	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓‘𝑆)𝑦) = (𝑥(+g‘𝑆)𝑦))
28	27	fveq2d 6867	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
29	8, 12	mhmf 18806	. . . . . . 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 7060	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑆) → (𝐹‘𝑥) ∈ (Base‘𝑇))
32	30	ffvelcdmi 7060	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑆) → (𝐹‘𝑦) ∈ (Base‘𝑇))
33	12, 24, 13	plusfval 18664	. . . . 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 605	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	26, 28, 34	3eqtr4d 2806	. . 3 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)))
36	17, 9	tmdcn 24123	. . . 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 34184	. 2 ⊢ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
39	13, 20	istmd 24114	. 2 ⊢ (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
40	3, 4, 38, 39	mpbir3an 1354	1 ⊢ 𝑇 ∈ TopMnd

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141   × cxp 5643  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  TopOpenctopn 17433  +_𝑓cplusf 18654  Mndcmnd 18751   MndHom cmhm 18798  TopOnctopon 22950  TopSpctps 22972   Cn ccn 23264   ×_t ctx 23600  Homeochmeo 23793  TopMndctmd 24110

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 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

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 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-map 8805  df-topgen 17455  df-plusf 18656  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-top 22934  df-topon 22951  df-topsp 22973  df-bases 22986  df-cn 23267  df-tx 23602  df-hmeo 23795  df-tmd 24112

This theorem is referenced by:  xrge0tmd  34203