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Theorem ghmcnp 24139
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 2735 . . . . . 6 𝐽 = 𝐽
21cnprcl 23269 . . . . 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 24105 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝑋))
763ad2ant1 1132 . . . . . . . 8 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝐽 ∈ (TopOn‘𝑋))
87adantr 480 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
9 simpl2 1191 . . . . . . . 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 2735 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
1210, 11tmdtopon 24105 . . . . . . . 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 484 . . . . . . 7 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
15 cnpf2 23274 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
168, 13, 14, 15syl3anc 1370 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋⟶(Base‘𝐻))
1716adantr 480 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹:𝑋⟶(Base‘𝐻))
1814adantr 480 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . 15 (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) = (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤))
2019mptpreima 6260 . . . . . . . . . . . . . 14 ((𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) “ 𝑦) = {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦}
219adantr 480 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ TopMnd)
2216adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹:𝑋⟶(Base‘𝐻))
23 simpll3 1213 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
24 ghmgrp1 19249 . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑥𝑋)
272adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 𝐽)
28 toponuni 22936 . . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . . 19 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
3130adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐴𝑋)
32 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (-g𝐺) = (-g𝐺)
335, 32grpsubcl 19051 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3425, 26, 31, 33syl3anc 1370 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
3522, 34ffvelcdmd 7105 . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
3719, 11, 36, 10tmdlactcn 24126 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ TopMnd ∧ (𝐹‘(𝑥(-g𝐺)𝐴)) ∈ (Base‘𝐻)) → (𝑤 ∈ (Base‘𝐻) ↦ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤)) ∈ (𝐾 Cn 𝐾))
3821, 35, 37syl2anc 584 . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝑦𝐾)
40 cnima 23289 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 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, 41eqeltrrid 2844 . . . . . . . . . . . . 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)
43 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑤 = (𝐹𝐴) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)))
4443eleq1d 2824 . . . . . . . . . . . . . 14 (𝑤 = (𝐹𝐴) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝐴)) ∈ 𝑦))
4522, 31ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝐴) ∈ (Base‘𝐻))
46 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (-g𝐻) = (-g𝐻)
475, 32, 46ghmsub 19255 . . . . . . . . . . . . . . . . . 18 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥𝑋𝐴𝑋) → (𝐹‘(𝑥(-g𝐺)𝐴)) = ((𝐹𝑥)(-g𝐻)(𝐹𝐴)))
4823, 26, 31, 47syl3anc 1370 . . . . . . . . . . . . . . . . 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)))
4948oveq1d 7446 . . . . . . . . . . . . . . . 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)))
50 ghmgrp2 19250 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐻 ∈ Grp)
5123, 50syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐻 ∈ Grp)
5222, 26ffvelcdmd 7105 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ (Base‘𝐻))
5311, 36, 46grpnpcan 19063 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐹𝑥) ∈ (Base‘𝐻) ∧ (𝐹𝐴) ∈ (Base‘𝐻)) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5451, 52, 45, 53syl3anc 1370 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (((𝐹𝑥)(-g𝐻)(𝐹𝐴))(+g𝐻)(𝐹𝐴)) = (𝐹𝑥))
5549, 54eqtrd 2775 . . . . . . . . . . . . . . 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))
56 simprrr 782 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → (𝐹𝑥) ∈ 𝑦)
5755, 56eqeltrd 2839 . . . . . . . . . . . . . 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)
5844, 45, 57elrabd 3697 . . . . . . . . . . . . 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})
59 cnpimaex 23280 . . . . . . . . . . . . 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}))
6018, 42, 58, 59syl3anc 1370 . . . . . . . . . . . 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}))
61 ssrab 4083 . . . . . . . . . . . . . . . 16 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} ↔ ((𝐹𝑧) ⊆ (Base‘𝐻) ∧ ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦))
6261simprbi 496 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ {𝑤 ∈ (Base‘𝐻) ∣ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦} → ∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦)
6322adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹:𝑋⟶(Base‘𝐻))
6463ffnd 6738 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝐹 Fn 𝑋)
658adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → 𝐽 ∈ (TopOn‘𝑋))
66 toponss 22949 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧𝐽) → 𝑧𝑋)
6765, 66sylan 580 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑧𝐽) → 𝑧𝑋)
68 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (𝑤 = (𝐹𝑣) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)))
6968eleq1d 2824 . . . . . . . . . . . . . . . . 17 (𝑤 = (𝐹𝑣) → (((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7069ralima 7257 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝑋𝑧𝑋) → (∀𝑤 ∈ (𝐹𝑧)((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)𝑤) ∈ 𝑦 ↔ ∀𝑣𝑧 ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) ∈ 𝑦))
7164, 67, 70syl2anc 584 . . . . . . . . . . . . . . 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))
7262, 71imbitrid 244 . . . . . . . . . . . . . 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))
73 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) = (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))
7473mptpreima 6260 . . . . . . . . . . . . . . . . 17 ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}
75 simpl1 1190 . . . . . . . . . . . . . . . . . . . 20 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐺 ∈ TopMnd)
7675ad2antrr 726 . . . . . . . . . . . . . . . . . . 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)
7725adantr 480 . . . . . . . . . . . . . . . . . . . 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)
7831adantr 480 . . . . . . . . . . . . . . . . . . . 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)
7926adantr 480 . . . . . . . . . . . . . . . . . . . 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)
805, 32grpsubcl 19051 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
8177, 78, 79, 80syl3anc 1370 . . . . . . . . . . . . . . . . . . 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)
82 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (+g𝐺) = (+g𝐺)
8373, 5, 82, 4tmdlactcn 24126 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ TopMnd ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋) → (𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽))
8476, 81, 83syl2anc 584 . . . . . . . . . . . . . . . . . 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))
85 simprl 771 . . . . . . . . . . . . . . . . . 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)
86 cnima 23289 . . . . . . . . . . . . . . . . . 18 (((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) ∈ (𝐽 Cn 𝐽) ∧ 𝑧𝐽) → ((𝑤𝑋 ↦ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)) “ 𝑧) ∈ 𝐽)
8784, 85, 86syl2anc 584 . . . . . . . . . . . . . . . . 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)
8874, 87eqeltrrid 2844 . . . . . . . . . . . . . . . 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)
89 oveq2 7439 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥))
9089eleq1d 2824 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 ↔ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) ∈ 𝑧))
915, 82, 32grpnpcan 19063 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ Grp ∧ 𝐴𝑋𝑥𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑥) = 𝐴)
9277, 78, 79, 91syl3anc 1370 . . . . . . . . . . . . . . . . . 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 781 . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . 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)
9590, 79, 94elrabd 3697 . . . . . . . . . . . . . . . 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})
96 simprrr 782 . . . . . . . . . . . . . . . . . . . . . 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)
97 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → (𝐹𝑣) = (𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤)))
9897oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑣 = ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) → ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹𝑣)) = ((𝐹‘(𝑥(-g𝐺)𝐴))(+g𝐻)(𝐹‘((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤))))
9998eleq1d 2824 . . . . . . . . . . . . . . . . . . . . . . 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))
10099rspccv 3619 . . . . . . . . . . . . . . . . . . . . . 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))
10196, 100syl 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))
102101adantr 480 . . . . . . . . . . . . . . . . . . . 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))
10323adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
10434adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝑥(-g𝐺)𝐴) ∈ 𝑋)
105103, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐺 ∈ Grp)
10631adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝐴𝑋)
10726adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑥𝑋)
108105, 106, 107, 80syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → (𝐴(-g𝐺)𝑥) ∈ 𝑋)
109 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → 𝑤𝑋)
1105, 82grpcl 18972 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ Grp ∧ (𝐴(-g𝐺)𝑥) ∈ 𝑋𝑤𝑋) → ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑋)
111105, 108, 109, 110syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . 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)
1125, 82, 36ghmlin 19252 . . . . . . . . . . . . . . . . . . . . . . . 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))))
113103, 104, 111, 112syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . 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))))
114 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (invg𝐺) = (invg𝐺)
1155, 32, 114grpinvsub 19053 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ 𝑥𝑋𝐴𝑋) → ((invg𝐺)‘(𝑥(-g𝐺)𝐴)) = (𝐴(-g𝐺)𝑥))
116105, 107, 106, 115syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
117116oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
118 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (0g𝐺) = (0g𝐺)
1195, 82, 118, 114grprinv 19021 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑥(-g𝐺)𝐴) ∈ 𝑋) → ((𝑥(-g𝐺)𝐴)(+g𝐺)((invg𝐺)‘(𝑥(-g𝐺)𝐴))) = (0g𝐺))
120105, 104, 119syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
121117, 120eqtr3d 2777 . . . . . . . . . . . . . . . . . . . . . . . . . 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))
122121oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . 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))
1235, 82grpass 18973 . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
124105, 104, 108, 109, 123syl13anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . 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)))
1255, 82, 118grplid 18998 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ Grp ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
126105, 109, 125syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) ∧ 𝑤𝑋) → ((0g𝐺)(+g𝐺)𝑤) = 𝑤)
127122, 124, 1263eqtr3d 2783 . . . . . . . . . . . . . . . . . . . . . . . 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)
128127fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . 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))
129113, 128eqtr3d 2777 . . . . . . . . . . . . . . . . . . . . . 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))
130129adantlr 715 . . . . . . . . . . . . . . . . . . . . 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))
131130eleq1d 2824 . . . . . . . . . . . . . . . . . . . 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))
132102, 131sylibd 239 . . . . . . . . . . . . . . . . . . 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))
133132ralrimiva 3144 . . . . . . . . . . . . . . . . . 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))
134 fveq2 6907 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑤 → (𝐹𝑣) = (𝐹𝑤))
135134eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑣 = 𝑤 → ((𝐹𝑣) ∈ 𝑦 ↔ (𝐹𝑤) ∈ 𝑦))
136135ralrab2 3707 . . . . . . . . . . . . . . . . . 18 (∀𝑣 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} (𝐹𝑣) ∈ 𝑦 ↔ ∀𝑤𝑋 (((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧 → (𝐹𝑤) ∈ 𝑦))
137133, 136sylibr 234 . . . . . . . . . . . . . . . . 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)
13822adantr 480 . . . . . . . . . . . . . . . . . . 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))
139138ffund 6741 . . . . . . . . . . . . . . . . . 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)
140 ssrab2 4090 . . . . . . . . . . . . . . . . . . 19 {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} ⊆ 𝑋
141138fdmd 6747 . . . . . . . . . . . . . . . . . . 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)
142140, 141sseqtrrid 4049 . . . . . . . . . . . . . . . . . 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)
143 funimass4 6973 . . . . . . . . . . . . . . . . . 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))
144139, 142, 143syl2anc 584 . . . . . . . . . . . . . . . . 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))
145137, 144mpbird 257 . . . . . . . . . . . . . . . 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)
146 eleq2 2828 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝑥𝑢𝑥 ∈ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
147 imaeq2 6076 . . . . . . . . . . . . . . . . . . 19 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → (𝐹𝑢) = (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}))
148147sseq1d 4027 . . . . . . . . . . . . . . . . . 18 (𝑢 = {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧} → ((𝐹𝑢) ⊆ 𝑦 ↔ (𝐹 “ {𝑤𝑋 ∣ ((𝐴(-g𝐺)𝑥)(+g𝐺)𝑤) ∈ 𝑧}) ⊆ 𝑦))
149146, 148anbi12d 632 . . . . . . . . . . . . . . . . 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)))
150149rspcev 3622 . . . . . . . . . . . . . . . 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))
15188, 95, 145, 150syl12anc 837 . . . . . . . . . . . . . . 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))
152151expr 456 . . . . . . . . . . . . . 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)))
15372, 152sylan2d 605 . . . . . . . . . . . . 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)))
154153rexlimdva 3153 . . . . . . . . . . . 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)))
15560, 154mpd 15 . . . . . . . . . . 11 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ (𝑥𝑋 ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦))) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
156155anassrs 467 . . . . . . . . . 10 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ (𝑦𝐾 ∧ (𝐹𝑥) ∈ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦))
157156expr 456 . . . . . . . . 9 (((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
158157ralrimiva 3144 . . . . . . . 8 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → ∀𝑦𝐾 ((𝐹𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ (𝐹𝑢) ⊆ 𝑦)))
1598adantr 480 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
16013adantr 480 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘(Base‘𝐻)))
161 simpr 484 . . . . . . . . 9 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝑥𝑋)
162 iscnp 23261 . . . . . . . . 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)))))
163159, 160, 161, 162syl3anc 1370 . . . . . . . 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)))))
16417, 158, 163mpbir2and 713 . . . . . . 7 ((((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) ∧ 𝑥𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
165164ralrimiva 3144 . . . . . 6 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))
166 cncnp 23304 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐻))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶(Base‘𝐻) ∧ ∀𝑥𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥))))
1678, 13, 166syl2anc 584 . . . . . 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))))
16816, 165, 167mpbir2and 713 . . . . 5 (((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ (𝐽 Cn 𝐾))
169168ex 412 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → 𝐹 ∈ (𝐽 Cn 𝐾)))
1703, 169jcad 512 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) → (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1711cncnpi 23302 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 𝐽) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
172171ancoms 458 . . 3 ((𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
173170, 172impbid1 225 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
1747, 28syl 17 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → 𝑋 = 𝐽)
175174eleq2d 2825 . . 3 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → (𝐴𝑋𝐴 𝐽))
176175anbi1d 631 . 2 ((𝐺 ∈ TopMnd ∧ 𝐻 ∈ TopMnd ∧ 𝐹 ∈ (𝐺 GrpHom 𝐻)) → ((𝐴𝑋𝐹 ∈ (𝐽 Cn 𝐾)) ↔ (𝐴 𝐽𝐹 ∈ (𝐽 Cn 𝐾))))
177173, 176bitr4d 282 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963   cuni 4912  cmpt 5231  ccnv 5688  dom cdm 5689  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  TopOpenctopn 17468  0gc0g 17486  Grpcgrp 18964  invgcminusg 18965  -gcsg 18966   GrpHom cghm 19243  TopOnctopon 22932   Cn ccn 23248   CnP ccnp 23249  TopMndctmd 24094
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-topgen 17490  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-ghm 19244  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096
This theorem is referenced by:  qqhcn  33954
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