Theorem mhmhmeotmd 33961

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 18698	. . 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 18697	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
7	1, 6	ax-mp 5	. . . 4 ⊢ 𝑆 ∈ Mnd
8		eqid 2733	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
9		eqid 2733	. . . . 5 ⊢ (+𝑓‘𝑆) = (+𝑓‘𝑆)
10	8, 9	mndplusf 18662	. . . 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 2733	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
13		eqid 2733	. . . . 5 ⊢ (+𝑓‘𝑇) = (+𝑓‘𝑇)
14	12, 13	mndplusf 18662	. . . 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 2733	. . . . 5 ⊢ (TopOpen‘𝑆) = (TopOpen‘𝑆)
18	17, 8	tmdtopon 23997	. . . 4 ⊢ (𝑆 ∈ TopMnd → (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆)))
19	16, 18	ax-mp 5	. . 3 ⊢ (TopOpen‘𝑆) ∈ (TopOn‘(Base‘𝑆))
20		eqid 2733	. . . . 5 ⊢ (TopOpen‘𝑇) = (TopOpen‘𝑇)
21	12, 20	istps 22850	. . . 4 ⊢ (𝑇 ∈ TopSp ↔ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇)))
22	4, 21	mpbi 230	. . 3 ⊢ (TopOpen‘𝑇) ∈ (TopOn‘(Base‘𝑇))
23		eqid 2733	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
24		eqid 2733	. . . . . 6 ⊢ (+g‘𝑇) = (+g‘𝑇)
25	8, 23, 24	mhmlin 18703	. . . . 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 1450	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g‘𝑆)𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
27	8, 23, 9	plusfval 18557	. . . . 5 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(+𝑓‘𝑆)𝑦) = (𝑥(+g‘𝑆)𝑦))
28	27	fveq2d 6832	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = (𝐹‘(𝑥(+g‘𝑆)𝑦)))
29	8, 12	mhmf 18699	. . . . . . 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 7022	. . . . 5 ⊢ (𝑥 ∈ (Base‘𝑆) → (𝐹‘𝑥) ∈ (Base‘𝑇))
32	30	ffvelcdmi 7022	. . . . 5 ⊢ (𝑦 ∈ (Base‘𝑆) → (𝐹‘𝑦) ∈ (Base‘𝑇))
33	12, 24, 13	plusfval 18557	. . . . 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 596	. . . 4 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)) = ((𝐹‘𝑥)(+g‘𝑇)(𝐹‘𝑦)))
35	26, 28, 34	3eqtr4d 2778	. . 3 ⊢ ((𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+𝑓‘𝑆)𝑦)) = ((𝐹‘𝑥)(+𝑓‘𝑇)(𝐹‘𝑦)))
36	17, 9	tmdcn 23999	. . . 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 33960	. 2 ⊢ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))
39	13, 20	istmd 23990	. 2 ⊢ (𝑇 ∈ TopMnd ↔ (𝑇 ∈ Mnd ∧ 𝑇 ∈ TopSp ∧ (+𝑓‘𝑇) ∈ (((TopOpen‘𝑇) ×t (TopOpen‘𝑇)) Cn (TopOpen‘𝑇))))
40	3, 4, 38, 39	mpbir3an 1342	1 ⊢ 𝑇 ∈ TopMnd

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113   × cxp 5617  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  +_gcplusg 17163  TopOpenctopn 17327  +_𝑓cplusf 18547  Mndcmnd 18644   MndHom cmhm 18691  TopOnctopon 22826  TopSpctps 22848   Cn ccn 23140   ×_t ctx 23476  Homeochmeo 23669  TopMndctmd 23986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-topgen 17349  df-plusf 18549  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cn 23143  df-tx 23478  df-hmeo 23671  df-tmd 23988

This theorem is referenced by:  xrge0tmd  33979