Theorem ghmcnp 24035

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 2729	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cnprcl 23165	. . . . 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 24001	. . . . . . . . 9 ⊢ (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
7	6	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
8	7	adantr 480	. . . . . . 7 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9		simpl2 1193	. . . . . . . 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 2729	. . . . . . . . 9 ⊢ (Base‘𝐻) = (Base‘𝐻)
12	10, 11	tmdtopon 24001	. . . . . . . 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 484	. . . . . . 7 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15		cnpf2 23170	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
16	8, 13, 14, 15	syl3anc 1373	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
17	16	adantr 480	. . . . . . . 8 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
18	14	adantr 480	. . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . 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 6199	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 1215	. . . . . . . . . . . . . . . . . . 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 19132	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ ∪ 𝐽)
28		toponuni 22834	. . . . . . . . . . . . . . . . . . . . 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 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ 𝑋)
31	30	adantr 480	. . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝐺) = (-g‘𝐺)
33	5, 32	grpsubcl 18934	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑥(-g‘𝐺)𝐴) ∈ 𝑋)
34	25, 26, 31, 33	syl3anc 1373	. . . . . . . . . . . . . . . . 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	ffvelcdmd 7039	. . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘𝐻) = (+g‘𝐻)
37	19, 11, 36, 10	tmdlactcn 24022	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 23185	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝐹‘𝐴) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝐴)))
44	43	eleq1d 2813	. . . . . . . . . . . . . 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	ffvelcdmd 7039	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . 19 ⊢ (-g‘𝐻) = (-g‘𝐻)
47	5, 32, 46	ghmsub 19138	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . 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 19133	. . . . . . . . . . . . . . . . . 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	ffvelcdmd 7039	. . . . . . . . . . . . . . . . 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 18946	. . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . 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 3658	. . . . . . . . . . . . 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 23176	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 4032	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 6671	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 22847	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
67	65, 66	sylan 580	. . . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = (𝐹‘𝑣) → ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)𝑤) = ((𝐹‘(𝑥(-g‘𝐺)𝐴))(+g‘𝐻)(𝐹‘𝑣)))
69	68	eleq1d 2813	. . . . . . . . . . . . . . . . 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 7193	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 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	imbitrid 244	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) = (𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤))
74	73	mptpreima 6199	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑤 ∈ 𝑋 ↦ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)) “ 𝑧) = {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧}
75		simpl1 1192	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
76	75	ad2antrr 726	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 18934	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝐴(-g‘𝐺)𝑥) ∈ 𝑋)
81	77, 78, 79, 80	syl3anc 1373	. . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . 20 ⊢ (+g‘𝐺) = (+g‘𝐺)
83	73, 5, 82, 4	tmdlactcn 24022	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . 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 23185	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . 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 7377	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝑥 → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥))
90	89	eleq1d 2813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) ∈ 𝑧))
91	5, 82, 32	grpnpcan 18946	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑥) = 𝐴)
92	77, 78, 79, 91	syl3anc 1373	. . . . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . 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 3658	. . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑣 = ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) → (𝐹‘𝑣) = (𝐹‘((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤)))
98	97	oveq2d 7385	. . . . . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . . . . 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 3582	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 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 18855	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐺 ∈ Grp ∧ (𝐴(-g‘𝐺)𝑥) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑋)
111	105, 108, 109, 110	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . 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 19135	. . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (invg‘𝐺) = (invg‘𝐺)
115	5, 32, 114	grpinvsub 18936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘(𝑥(-g‘𝐺)𝐴)) = (𝐴(-g‘𝐺)𝑥))
116	105, 107, 106, 115	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (0g‘𝐺) = (0g‘𝐺)
119	5, 82, 118, 114	grprinv 18904	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . . . . . 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 18856	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . . . . . 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 18881	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐺 ∈ Grp ∧ 𝑤 ∈ 𝑋) → ((0g‘𝐺)(+g‘𝐺)𝑤) = 𝑤)
126	105, 109, 125	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . . . . . . 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 6840	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑤 → (𝐹‘𝑣) = (𝐹‘𝑤))
135	134	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 = 𝑤 → ((𝐹‘𝑣) ∈ 𝑦 ↔ (𝐹‘𝑤) ∈ 𝑦))
136	135	ralrab2 3666	. . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 6674	. . . . . . . . . . . . . . . . . 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 4039	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑤 ∈ 𝑋 ∣ ((𝐴(-g‘𝐺)𝑥)(+g‘𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
141	138	fdmd 6680	. . . . . . . . . . . . . . . . . . 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 3987	. . . . . . . . . . . . . . . . . 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 6907	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . . . . 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 6016	. . . . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . . . 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 3585	. . . . . . . . . . . . . . . 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 836	. . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . 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 605	. . . . . . . . . . . . 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 3134	. . . . . . . . . . . 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 467	. . . . . . . . . 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 456	. . . . . . . . 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 3125	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
160	13	adantr 480	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
161		simpr 484	. . . . . . . . 9 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
162		iscnp 23157	. . . . . . . . 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 1373	. . . . . . . 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 713	. . . . . . 7 ⊢ ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
165	164	ralrimiva 3125	. . . . . 6 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
166		cncnp 23200	. . . . . . 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 584	. . . . . 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 713	. . . . 5 ⊢ (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
169	168	ex 412	. . . 4 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
170	3, 169	jcad 512	. . 3 ⊢ ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
171	1	cncnpi 23198	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ ∪ 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
172	171	ancoms 458	. . 3 ⊢ ((𝐴 ∈ ∪ 𝐽 ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
173	170, 172	impbid1 225	. 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 2814	. . 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 282	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3402   ⊆ wss 3911  ∪ cuni 4867   ↦ cmpt 5183  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  +_gcplusg 17196  TopOpenctopn 17360  0_gc0g 17378  Grpcgrp 18847  inv_gcminusg 18848  -_gcsg 18849   GrpHom cghm 19126  TopOnctopon 22830   Cn ccn 23144   CnP ccnp 23145  TopMndctmd 23990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-0g 17380  df-topgen 17382  df-plusf 18548  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-grp 18850  df-minusg 18851  df-sbg 18852  df-ghm 19127  df-top 22814  df-topon 22831  df-topsp 22853  df-bases 22866  df-cn 23147  df-cnp 23148  df-tx 23482  df-tmd 23992

This theorem is referenced by:  qqhcn  33974