Theorem ghmcnp 22717

Description: A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)

Hypotheses

Ref	Expression
ghmcnp.x	⊢ 𝑋 = (Base‘𝐺)
ghmcnp.j	⊢ 𝐽 = (TopOpen‘𝐺)
ghmcnp.k	⊢ 𝐾 = (TopOpen‘𝐻)

Assertion

Ref	Expression
ghmcnp	⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Proof of Theorem ghmcnp

Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2821	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cnprcl 21847	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐴 ∈ ∪ 𝐽)
3	2	a1i 11	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐴 ∈ ∪ 𝐽))
4		ghmcnp.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝐺)
5		ghmcnp.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝐺)
6	4, 5	tmdtopon 22683	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
7	6	3ad2ant1 1129	. . . . . . . 8 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 483	. . . . . . 7 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9		simpl2 1188	. . . . . . . 8 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐻 ∈ TopMnd)
10		ghmcnp.k	. . . . . . . . 9 ⊢ 𝐾 = (TopOpen‘𝐻)
11		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
12	10, 11	tmdtopon 22683	. . . . . . . 8 ⊢ (𝐻 ∈ TopMnd → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
13	9, 12	syl 17	. . . . . . 7 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
14		simpr 487	. . . . . . 7 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15		cnpf2 21852	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
16	8, 13, 14, 15	syl3anc 1367	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
17	16	adantr 483	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
18	14	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
19		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤))
20	19	mptpreima 6086	. . . . . . . . . . . . . 14 ⊢ (◡(𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}
21	9	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
22	16	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
23		simpll3 1210	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24		ghmgrp1 18354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
25	23, 24	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐺 ∈ Grp)
26		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝑥 ∈ 𝑋)
27	2	adantl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
28		toponuni 21516	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
29	8, 28	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝑋 = ∪ 𝐽)
30	27, 29	eleqtrrd 2916	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
31	30	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐴 ∈ 𝑋)
32		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝐺) = (-g‘𝐺)
33	5, 32	grpsubcl 18173	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑋)
34	25, 26, 31, 33	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑋)
35	22, 34	ffvelrnd 6846	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g‘𝐺)𝐴)) ∈ (Base‘𝐻))
36		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐻) = (+g‘𝐻)
37	19, 11, 36, 10	tmdlactcn 22704	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g‘𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
38	21, 35, 37	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
39		simprrl 779	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝑦 ∈ 𝐾)
40		cnima 21867	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) ∈ (𝐾 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡(𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
41	38, 39, 40	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (◡(𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
42	20, 41	eqeltrrid 2918	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦} ∈ 𝐾)
43		oveq2 7158	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐹‘𝐴) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)))
44	43	eleq1d 2897	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝐹‘𝐴) → (((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)) ∈ 𝑦))
45	22, 31	ffvelrnd 6846	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘𝐴) ∈ (Base‘𝐻))
46		eqid 2821	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝐻) = (-g‘𝐻)
47	5, 32, 46	ghmsub 18360	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘(𝑥(-g‘𝐺)𝐴)) = ((𝐹‘𝑥)(-g‘𝐻)(𝐹‘𝐴)))
48	23, 26, 31, 47	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g‘𝐺)𝐴)) = ((𝐹‘𝑥)(-g‘𝐻)(𝐹‘𝐴)))
49	48	oveq1d 7165	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)) = (((𝐹‘𝑥)(-g‘𝐻)(𝐹‘𝐴))(+g‘𝐻)(𝐹‘𝐴)))
50		ghmgrp2 18355	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
51	23, 50	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐻 ∈ Grp)
52	22, 26	ffvelrnd 6846	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘𝑥) ∈ (Base‘𝐻))
53	11, 36, 46	grpnpcan 18185	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻 ∈ Grp ∧ (𝐹‘𝑥) ∈ (Base‘𝐻) ∧ (𝐹‘𝐴) ∈ (Base‘𝐻)) → (((𝐹‘𝑥)(-g‘𝐻)(𝐹‘𝐴))(+g‘𝐻)(𝐹‘𝐴)) = (𝐹‘𝑥))
54	51, 52, 45, 53	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (((𝐹‘𝑥)(-g‘𝐻)(𝐹‘𝐴))(+g‘𝐻)(𝐹‘𝐴)) = (𝐹‘𝑥))
55	49, 54	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)) = (𝐹‘𝑥))
56		simprrr 780	. . . . . . . . . . . . . . 15 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘𝑥) ∈ 𝑦)
57	55, 56	eqeltrd 2913	. . . . . . . . . . . . . 14 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)) ∈ 𝑦)
58	44, 45, 57	elrabd 3681	. . . . . . . . . . . . 13 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (𝐹‘𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦})
59		cnpimaex 21858	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦} ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}) → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}))
60	18, 42, 58, 59	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → ∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}))
61		ssrab 4048	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹 “ 𝑧) ⊆ (Base‘𝐻) ∧ ∀𝑤 ∈ (𝐹 “ 𝑧)((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦))
62	61	simprbi 499	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹 “ 𝑧)((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦)
63	22	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
64	63	ffnd 6509	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → 𝐹 Fn 𝑋)
65	8	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
66		toponss 21529	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
67	65, 66	sylan 582	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
68		oveq2 7158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝐹‘𝑣) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)))
69	68	eleq1d 2897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = (𝐹‘𝑣) → (((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))
70	69	ralima 6994	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑧 ⊆ 𝑋) → (∀𝑤 ∈ (𝐹 “ 𝑧)((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))
71	64, 67, 70	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → (∀𝑤 ∈ (𝐹 “ 𝑧)((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))
72	62, 71	syl5ib 246	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → ((𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦} → ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))
73		eqid 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))
74	73	mptpreima 6086	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) “ 𝑧) = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}
75		simpl1 1187	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
76	75	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝐺 ∈ TopMnd)
77	25	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝐺 ∈ Grp)
78	31	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝐴 ∈ 𝑋)
79	26	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝑥 ∈ 𝑋)
80	5, 32	grpsubcl 18173	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐴(-g‘𝐺)𝑥) ∈ 𝑋)
81	77, 78, 79, 80	syl3anc 1367	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → (𝐴(-g‘𝐺)𝑥) ∈ 𝑋)
82		eqid 2821	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+g‘𝐺) = (+g‘𝐺)
83	73, 5, 82, 4	tmdlactcn 22704	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ TopMnd ∧ (𝐴(-g‘𝐺)𝑥) ∈ 𝑋) → (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
84	76, 81, 83	syl2anc 586	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
85		simprl 769	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝑧 ∈ 𝐽)
86		cnima 21867	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧 ∈ 𝐽) → (◡(𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
87	84, 85, 86	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → (◡(𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
88	74, 87	eqeltrrid 2918	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ∈ 𝐽)
89		oveq2 7158	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑥 → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥))
90	89	eleq1d 2897	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧))
91	5, 82, 32	grpnpcan 18185	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) = 𝐴)
92	77, 78, 79, 91	syl3anc 1367	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) = 𝐴)
93		simprrl 779	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝐴 ∈ 𝑧)
94	92, 93	eqeltrd 2913	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧)
95	90, 79, 94	elrabd 3681	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝑥 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧})
96		simprrr 780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦)
97		fveq2 6664	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) → (𝐹‘𝑣) = (𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)))
98	97	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑣 = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))))
99	98	eleq1d 2897	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑣 = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) → (((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) ∈ 𝑦))
100	99	rspccv 3619	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦 → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) ∈ 𝑦))
101	96, 100	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) ∈ 𝑦))
102	101	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) ∈ 𝑦))
103	23	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
104	34	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑋)
105	103, 24	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → 𝐺 ∈ Grp)
106	31	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → 𝐴 ∈ 𝑋)
107	26	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → 𝑥 ∈ 𝑋)
108	105, 106, 107, 80	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (𝐴(-g‘𝐺)𝑥) ∈ 𝑋)
109		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑋)
110	5, 82	grpcl 18105	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ Grp ∧ (𝐴(-g‘𝐺)𝑥) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑋)
111	105, 108, 109, 110	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑋)
112	5, 82, 36	ghmlin 18357	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑥(-g‘𝐺)𝐴) ∈ 𝑋 ∧ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑋) → (𝐹‘((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))))
113	103, 104, 111, 112	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (𝐹‘((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))))
114		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (invg‘𝐺) = (invg‘𝐺)
115	5, 32, 114	grpinvsub 18175	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴)) = (𝐴(-g‘𝐺)𝑥))
116	105, 107, 106, 115	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴)) = (𝐴(-g‘𝐺)𝑥))
117	116	oveq2d 7166	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴))) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)(𝐴(-g‘𝐺)𝑥)))
118		eqid 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0g‘𝐺) = (0g‘𝐺)
119	5, 82, 118, 114	grprinv 18147	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ Grp ∧ (𝑥(-g‘𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴))) = (0g‘𝐺))
120	105, 104, 119	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴))) = (0g‘𝐺))
121	117, 120	eqtr3d 2858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)(𝐴(-g‘𝐺)𝑥)) = (0g‘𝐺))
122	121	oveq1d 7165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)(𝐴(-g‘𝐺)𝑥))(+g‘𝐺)𝑤) = ((0g‘𝐺)(+g‘𝐺)𝑤))
123	5, 82	grpass 18106	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ Grp ∧ ((𝑥(-g‘𝐺)𝐴) ∈ 𝑋 ∧ (𝐴(-g‘𝐺)𝑥) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)(𝐴(-g‘𝐺)𝑥))(+g‘𝐺)𝑤) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)))
124	105, 104, 108, 109, 123	syl13anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)(𝐴(-g‘𝐺)𝑥))(+g‘𝐺)𝑤) = ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)))
125	5, 82, 118	grplid 18127	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑤) = 𝑤)
126	105, 109, 125	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑤) = 𝑤)
127	122, 124, 126	3eqtr3d 2864	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) = 𝑤)
128	127	fveq2d 6668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (𝐹‘((𝑥(-g‘𝐺)𝐴)(+g‘𝐺)((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) = (𝐹‘𝑤))
129	113, 128	eqtr3d 2858	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) = (𝐹‘𝑤))
130	129	adantlr 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) = (𝐹‘𝑤))
131	130	eleq1d 2897	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))) ∈ 𝑦 ↔ (𝐹‘𝑤) ∈ 𝑦))
132	102, 131	sylibd 241	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) ∧ 𝑤 ∈ 𝑋) → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑦))
133	132	ralrimiva 3182	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ∀𝑤 ∈ 𝑋 (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑦))
134		fveq2 6664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝐹‘𝑣) = (𝐹‘𝑤))
135	134	eleq1d 2897	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝐹‘𝑣) ∈ 𝑦 ↔ (𝐹‘𝑤) ∈ 𝑦))
136	135	ralrab2 3689	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑣 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} (𝐹‘𝑣) ∈ 𝑦 ↔ ∀𝑤 ∈ 𝑋 (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 → (𝐹‘𝑤) ∈ 𝑦))
137	133, 136	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ∀𝑣 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} (𝐹‘𝑣) ∈ 𝑦)
138	22	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
139	138	ffund 6512	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → Fun 𝐹)
140		ssrab2 4055	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
141	138	fdmd 6517	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → dom 𝐹 = 𝑋)
142	140, 141	sseqtrrid 4019	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹)
143		funimass4 6724	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun 𝐹 ∧ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹) → ((𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} (𝐹‘𝑣) ∈ 𝑦))
144	139, 142, 143	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ((𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} (𝐹‘𝑣) ∈ 𝑦))
145	137, 144	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → (𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)
146		eleq2 2901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}))
147		imaeq2 5919	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} → (𝐹 “ 𝑢) = (𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}))
148	147	sseq1d 3997	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} → ((𝐹 “ 𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
149	146, 148	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} → ((𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)))
150	149	rspcev 3622	. . . . . . . . . . . . . . . 16 ⊢ (({𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ∈ 𝐽 ∧ (𝑥 ∈ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
151	88, 95, 145, 150	syl12anc 834	. . . . . . . . . . . . . . 15 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ (𝑧 ∈ 𝐽 ∧ (𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦))) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
152	151	expr 459	. . . . . . . . . . . . . 14 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → ((𝐴 ∈ 𝑧 ∧ ∀𝑣 ∈ 𝑧 ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)) ∈ 𝑦) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))
153	72, 152	sylan2d 606	. . . . . . . . . . . . 13 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) ∧ 𝑧 ∈ 𝐽) → ((𝐴 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))
154	153	rexlimdva 3284	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → (∃𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 ∧ (𝐹 “ 𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) ∈ 𝑦}) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))
155	60, 154	mpd 15	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦))) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
156	155	anassrs 470	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝑥) ∈ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦))
157	156	expr 459	. . . . . . . . 9 ⊢ (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))
158	157	ralrimiva 3182	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))
159	8	adantr 483	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
160	13	adantr 483	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
161		simpr 487	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
162		iscnp 21839	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))))
163	159, 160, 161, 162	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ (𝐹 “ 𝑢) ⊆ 𝑦)))))
164	17, 158, 163	mpbir2and 711	. . . . . . 7 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
165	164	ralrimiva 3182	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
166		cncnp 21882	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
167	8, 13, 166	syl2anc 586	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
168	16, 165, 167	mpbir2and 711	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
169	168	ex 415	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
170	3, 169	jcad 515	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
171	1	cncnpi 21880	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
172	171	ancoms 461	. . 3 ⊢ ((𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
173	170, 172	impbid1 227	. 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
174	7, 28	syl 17	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝑋 = ∪ 𝐽)
175	174	eleq2d 2898	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ 𝐽))
176	175	anbi1d 631	. 2 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
177	173, 176	bitr4d 284	1 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 ∈ 𝑋 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142   ⊆ wss 3935  ∪ cuni 4831   ↦ cmpt 5138  ^◡ccnv 5548  dom cdm 5549   “ cima 5552  Fun wfun 6343   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  TopOpenctopn 16689  0_gc0g 16707  Grpcgrp 18097  inv_gcminusg 18098  -_gcsg 18099   GrpHom cghm 18349  TopOnctopon 21512   Cn ccn 21826   CnP ccnp 21827  TopMndctmd 22672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-map 8402  df-0g 16709  df-topgen 16711  df-plusf 17845  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-sbg 18102  df-ghm 18350  df-top 21496  df-topon 21513  df-topsp 21535  df-bases 21548  df-cn 21829  df-cnp 21830  df-tx 22164  df-tmd 22674

This theorem is referenced by:  qqhcn  31227