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Theorem ghmcnp 21831
 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) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))

Proof of Theorem ghmcnp
Dummy variables 𝑣 𝑢 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 𝐽 = 𝐽
21cnprcl 20962 . . . . 5 (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐴 𝐽)
32a1i 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)
64, 5tmdtopon 21798 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
763ad2ant1 1080 . . . . . . . 8 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
87adantr 481 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9 simpl2 1063 . . . . . . . 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 2621 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
1210, 11tmdtopon 21798 . . . . . . . 8 (𝐻 ∈ TopMnd → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
139, 12syl 17 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
14 simpr 477 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15 cnpf2 20967 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
168, 13, 14, 15syl3anc 1323 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
1716adantr 481 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
1814adantr 481 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
19 eqid 2621 . . . . . . . . . . . . . . 15 (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤))
2019mptpreima 5589 . . . . . . . . . . . . . 14 ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}
219adantr 481 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
2216adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
23 simpll3 1100 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24 ghmgrp1 17586 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
2523, 24syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐺 ∈ Grp)
26 simprl 793 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑥𝑋)
272adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
28 toponuni 20641 . . . . . . . . . . . . . . . . . . . . 21 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
298, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝑋 = 𝐽)
3027, 29eleqtrrd 2701 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
3130adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐴𝑋)
32 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (-g𝐺) = (-g𝐺)
335, 32grpsubcl 17419 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3425, 26, 31, 33syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3522, 34ffvelrnd 6318 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻))
36 eqid 2621 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
3719, 11, 36, 10tmdlactcn 21819 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
3821, 35, 37syl2anc 692 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
39 simprrl 803 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑦𝐾)
40 cnima 20982 . . . . . . . . . . . . . . 15 (((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾) ∧ 𝑦𝐾) → ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
4138, 39, 40syl2anc 692 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) ∈ 𝐾)
4220, 41syl5eqelr 2703 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ∈ 𝐾)
4322, 31ffvelrnd 6318 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ (Base‘𝐻))
44 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (-g𝐻) = (-g𝐻)
455, 32, 44ghmsub 17592 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋𝐴𝑋) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4623, 26, 31, 45syl3anc 1323 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4746oveq1d 6622 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) = (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)))
48 ghmgrp2 17587 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
4923, 48syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ Grp)
5022, 26ffvelrnd 6318 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ (Base‘𝐻))
5111, 36, 44grpnpcan 17431 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐻) ∧ (𝐹𝐴) ∈ (Base‘𝐻)) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5249, 50, 43, 51syl3anc 1323 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5347, 52eqtrd 2655 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
54 simprrr 804 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ 𝑦)
5553, 54eqeltrd 2698 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦)
56 oveq2 6615 . . . . . . . . . . . . . . . 16 (𝑤 = (𝐹𝐴) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)))
5756eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝐴) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
5857elrab 3347 . . . . . . . . . . . . . 14 ((𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝐴) ∈ (Base‘𝐻) ∧ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
5943, 55, 58sylanbrc 697 . . . . . . . . . . . . 13 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦})
60 cnpimaex 20973 . . . . . . . . . . . . 13 ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ∧ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ∈ 𝐾 ∧ (𝐹𝐴) ∈ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}))
6118, 42, 59, 60syl3anc 1323 . . . . . . . . . . . 12 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}))
62 ssrab 3661 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝑧) ⊆ (Base‘𝐻) ∧ ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦))
6362simprbi 480 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦)
6422adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
65 ffn 6004 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋⟶(Base‘𝐻) → 𝐹 Fn 𝑋)
6664, 65syl 17 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹 Fn 𝑋)
678adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
68 toponss 20643 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
6967, 68sylan 488 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝑧𝑋)
70 oveq2 6615 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹𝑣) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)))
7170eleq1d 2683 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹𝑣) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7271ralima 6455 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑋𝑧𝑋) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7366, 69, 72syl2anc 692 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7463, 73syl5ib 234 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
75 eqid 2621 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))
7675mptpreima 5589 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}
77 simpl1 1062 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
7877ad2antrr 761 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐺 ∈ TopMnd)
7925adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐺 ∈ Grp)
8031adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐴𝑋)
8126adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑥𝑋)
825, 32grpsubcl 17419 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
8379, 80, 81, 82syl3anc 1323 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
84 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (+g𝐺) = (+g𝐺)
8575, 5, 84, 4tmdlactcn 21819 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ TopMnd ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
8678, 83, 85syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
87 simprl 793 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑧𝐽)
88 cnima 20982 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8986, 87, 88syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
9076, 89syl5eqelr 2703 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∈ 𝐽)
915, 84, 32grpnpcan 17431 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
9279, 80, 81, 91syl3anc 1323 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
93 simprrl 803 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐴𝑧)
9492, 93eqeltrd 2698 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧)
95 oveq2 6615 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥))
9695eleq1d 2683 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
9796elrab 3347 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ↔ (𝑥𝑋 ∧ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
9881, 94, 97sylanbrc 697 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧})
99 simprrr 804 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦)
100 fveq2 6150 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (𝐹𝑣) = (𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
101100oveq2d 6623 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
102101eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
103102rspccv 3292 . . . . . . . . . . . . . . . . . . . . . 22 (∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
10499, 103syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
105104adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦))
10623adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10734adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
108106, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐺 ∈ Grp)
10931adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐴𝑋)
11026adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑥𝑋)
111108, 109, 110, 82syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
112 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑤𝑋)
1135, 84grpcl 17354 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Grp ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
114108, 111, 112, 113syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
1155, 84, 36ghmlin 17589 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋 ∧ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
116106, 107, 114, 115syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
117 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (invg𝐺) = (invg𝐺)
1185, 32, 117grpinvsub 17421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
119108, 110, 109, 118syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
120119oveq2d 6623 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = ((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥)))
121 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0g𝐺) = (0g𝐺)
1225, 84, 121, 117grprinv 17393 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
123108, 107, 122syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
124120, 123eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥)) = (0g𝐺))
125124oveq1d 6622 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((0g𝐺)(+g𝐺)𝑤))
1265, 84grpass 17355 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ ((𝑥(-g𝐺)𝐴) ∈ 𝑋 ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋)) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
127108, 107, 111, 112, 126syl13anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝑥(-g𝐺)𝐴)(+g𝐺)(𝐴(-g𝐺)𝑥))(+g𝐺)𝑤) = ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
1285, 84, 121grplid 17376 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
129108, 112, 128syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
130125, 127, 1293eqtr3d 2663 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = 𝑤)
131130fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐹‘((𝑥(-g𝐺)𝐴)(+g𝐺)((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
132116, 131eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
133132adantlr 750 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) = (𝐹𝑤))
134133eleq1d 2683 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
135105, 134sylibd 229 . . . . . . . . . . . . . . . . . . 19 ((((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) ∧ 𝑤𝑋) → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
136135ralrimiva 2960 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
137 fveq2 6150 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
138137eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → ((𝐹𝑣) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
139138ralrab2 3355 . . . . . . . . . . . . . . . . . 18 (∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦 ↔ ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
140136, 139sylibr 224 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦)
14122adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
142 ffun 6007 . . . . . . . . . . . . . . . . . . 19 (𝐹:𝑋⟶(Base‘𝐻) → Fun 𝐹)
143141, 142syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → Fun 𝐹)
144 ssrab2 3668 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
145 fdm 6010 . . . . . . . . . . . . . . . . . . . 20 (𝐹:𝑋⟶(Base‘𝐻) → dom 𝐹 = 𝑋)
146141, 145syl 17 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → dom 𝐹 = 𝑋)
147144, 146syl5sseqr 3635 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹)
148 funimass4 6206 . . . . . . . . . . . . . . . . . 18 ((Fun 𝐹 ∧ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ dom 𝐹) → ((𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦))
149143, 147, 148syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ((𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦 ↔ ∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦))
150140, 149mpbird 247 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)
151 eleq2 2687 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝑥𝑢𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
152 imaeq2 5423 . . . . . . . . . . . . . . . . . . 19 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝐹𝑢) = (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
153152sseq1d 3613 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝐹𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
154151, 153anbi12d 746 . . . . . . . . . . . . . . . . 17 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)))
155154rspcev 3295 . . . . . . . . . . . . . . . 16 (({𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∈ 𝐽 ∧ (𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ∧ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
15690, 98, 150, 155syl12anc 1321 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ (𝑧𝐽 ∧ (𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
157156expr 642 . . . . . . . . . . . . . 14 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐴𝑧 ∧ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
15874, 157sylan2d 499 . . . . . . . . . . . . 13 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → ((𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
159158rexlimdva 3024 . . . . . . . . . . . 12 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (∃𝑧𝐽 (𝐴𝑧 ∧ (𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
16061, 159mpd 15 . . . . . . . . . . 11 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
161160anassrs 679 . . . . . . . . . 10 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
162161expr 642 . . . . . . . . 9 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
163162ralrimiva 2960 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
1648adantr 481 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
16513adantr 481 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
166 simpr 477 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝑥𝑋)
167 iscnp 20954 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))))
168164, 165, 166, 167syl3anc 1323 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))))
16917, 163, 168mpbir2and 956 . . . . . . 7 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
170169ralrimiva 2960 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
171 cncnp 20997 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
1728, 13, 171syl2anc 692 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
17316, 170, 172mpbir2and 956 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
174173ex 450 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
1753, 174jcad 555 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1761cncnpi 20995 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
177176ancoms 469 . . 3 ((𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
178175, 177impbid1 215 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1797, 28syl 17 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝑋 = 𝐽)
180179eleq2d 2684 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴𝑋𝐴 𝐽))
181180anbi1d 740 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
182178, 181bitr4d 271 1 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  {crab 2911   ⊆ wss 3556  ∪ cuni 4404   ↦ cmpt 4675  ◡ccnv 5075  dom cdm 5076   “ cima 5079  Fun wfun 5843   Fn wfn 5844  ⟶wf 5845  ‘cfv 5849  (class class class)co 6607  Basecbs 15784  +gcplusg 15865  TopOpenctopn 16006  0gc0g 16024  Grpcgrp 17346  invgcminusg 17347  -gcsg 17348   GrpHom cghm 17581  TopOnctopon 20637   Cn ccn 20941   CnP ccnp 20942  TopMndctmd 21787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807  df-0g 16026  df-topgen 16028  df-plusf 17165  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-ghm 17582  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-cn 20944  df-cnp 20945  df-tx 21278  df-tmd 21789 This theorem is referenced by:  qqhcn  29829
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