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Theorem ghmquskerlem3 19231
Description: The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypotheses
Ref Expression
ghmqusker.1 0 = (0g𝐻)
ghmqusker.f (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusker.k 𝐾 = (𝐹 “ { 0 })
ghmqusker.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
ghmqusker.j 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
Assertion
Ref Expression
ghmquskerlem3 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑄,𝑞   𝜑,𝑞
Allowed substitution hint:   0 (𝑞)

Proof of Theorem ghmquskerlem3
Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . 2 (Base‘𝑄) = (Base‘𝑄)
2 eqid 2728 . 2 (Base‘𝐻) = (Base‘𝐻)
3 eqid 2728 . 2 (+g𝑄) = (+g𝑄)
4 eqid 2728 . 2 (+g𝐻) = (+g𝐻)
5 ghmqusker.k . . . 4 𝐾 = (𝐹 “ { 0 })
6 ghmqusker.f . . . . 5 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
7 ghmqusker.1 . . . . . 6 0 = (0g𝐻)
87ghmker 19190 . . . . 5 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
96, 8syl 17 . . . 4 (𝜑 → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
105, 9eqeltrid 2833 . . 3 (𝜑𝐾 ∈ (NrmSGrp‘𝐺))
11 ghmqusker.q . . . 4 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
1211qusgrp 19135 . . 3 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
1310, 12syl 17 . 2 (𝜑𝑄 ∈ Grp)
14 ghmrn 19177 . . 3 (𝐹 ∈ (𝐺 GrpHom 𝐻) → ran 𝐹 ∈ (SubGrp‘𝐻))
15 subgrcl 19080 . . 3 (ran 𝐹 ∈ (SubGrp‘𝐻) → 𝐻 ∈ Grp)
166, 14, 153syl 18 . 2 (𝜑𝐻 ∈ Grp)
176adantr 480 . . . . 5 ((𝜑𝑞 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
1817imaexd 7919 . . . 4 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
1918uniexd 7742 . . 3 ((𝜑𝑞 ∈ (Base‘𝑄)) → (𝐹𝑞) ∈ V)
20 ghmqusker.j . . . 4 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
2120a1i 11 . . 3 (𝜑𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞)))
22 simpr 484 . . . . 5 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) = (𝐹𝑥))
23 eqid 2728 . . . . . . . . . 10 (Base‘𝐺) = (Base‘𝐺)
2423, 2ghmf 19168 . . . . . . . . 9 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
256, 24syl 17 . . . . . . . 8 (𝜑𝐹:(Base‘𝐺)⟶(Base‘𝐻))
2625frnd 6725 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (Base‘𝐻))
2726ad3antrrr 729 . . . . . 6 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ran 𝐹 ⊆ (Base‘𝐻))
2825ffnd 6718 . . . . . . . 8 (𝜑𝐹 Fn (Base‘𝐺))
2928ad3antrrr 729 . . . . . . 7 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 Fn (Base‘𝐺))
3011a1i 11 . . . . . . . . . . . . 13 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
31 eqidd 2729 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
32 ovexd 7450 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
33 ghmgrp1 19166 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ Grp)
346, 33syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
3530, 31, 32, 34qusbas 17521 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
36 nsgsubg 19107 . . . . . . . . . . . . . 14 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
37 eqid 2728 . . . . . . . . . . . . . . 15 (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
3823, 37eqger 19127 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
3910, 36, 383syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
4039qsss 8791 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
4135, 40eqsstrrd 4018 . . . . . . . . . . 11 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
4241sselda 3979 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
4342elpwid 4608 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
4443sselda 3979 . . . . . . . 8 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) → 𝑥 ∈ (Base‘𝐺))
4544adantr 480 . . . . . . 7 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑥 ∈ (Base‘𝐺))
4629, 45fnfvelrnd 7087 . . . . . 6 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ ran 𝐹)
4727, 46sseldd 3980 . . . . 5 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐹𝑥) ∈ (Base‘𝐻))
4822, 47eqeltrd 2829 . . . 4 ((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽𝑟) ∈ (Base‘𝐻))
496adantr 480 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
50 simpr 484 . . . . 5 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
517, 49, 5, 11, 20, 50ghmquskerlem2 19230 . . . 4 ((𝜑𝑟 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
5248, 51r19.29a 3158 . . 3 ((𝜑𝑟 ∈ (Base‘𝑄)) → (𝐽𝑟) ∈ (Base‘𝐻))
5319, 21, 52fmpt2d 7129 . 2 (𝜑𝐽:(Base‘𝑄)⟶(Base‘𝐻))
5439ad6antr 735 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
5550ad5antr 733 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ (Base‘𝑄))
5635ad6antr 735 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
5755, 56eleqtrrd 2832 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
58 simp-4r 783 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥𝑟)
59 qsel 8809 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
6054, 57, 58, 59syl3anc 1369 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
61 simp-5r 785 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ (Base‘𝑄))
6261, 56eleqtrrd 2832 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
63 simplr 768 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦𝑠)
64 qsel 8809 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑦𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
6554, 62, 63, 64syl3anc 1369 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
6660, 65oveq12d 7433 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(+g𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)))
6710ad6antr 735 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐾 ∈ (NrmSGrp‘𝐺))
6843ad5antr 733 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ⊆ (Base‘𝐺))
6968, 58sseldd 3980 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥 ∈ (Base‘𝐺))
7041sselda 3979 . . . . . . . . . . . . . 14 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
7170elpwid 4608 . . . . . . . . . . . . 13 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
7271adantlr 714 . . . . . . . . . . . 12 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
7372ad4antr 731 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ⊆ (Base‘𝐺))
7473, 63sseldd 3980 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦 ∈ (Base‘𝐺))
75 eqid 2728 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
7611, 23, 75, 3qusadd 19137 . . . . . . . . . 10 ((𝐾 ∈ (NrmSGrp‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7767, 69, 74, 76syl3anc 1369 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ([𝑥](𝐺 ~QG 𝐾)(+g𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7866, 77eqtrd 2768 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(+g𝑄)𝑠) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾))
7978fveq2d 6896 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = (𝐽‘[(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾)))
806ad6antr 735 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8180, 33syl 17 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ Grp)
8223, 75, 81, 69, 74grpcld 18898 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑥(+g𝐺)𝑦) ∈ (Base‘𝐺))
837, 80, 5, 11, 20, 82ghmquskerlem1 19228 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝐾)) = (𝐹‘(𝑥(+g𝐺)𝑦)))
8423, 75, 4ghmlin 19169 . . . . . . . 8 ((𝐹 ∈ (𝐺 GrpHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(+g𝐺)𝑦)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
8580, 69, 74, 84syl3anc 1369 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐹‘(𝑥(+g𝐺)𝑦)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
8679, 83, 853eqtrd 2772 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
87 simpllr 775 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑟) = (𝐹𝑥))
88 simpr 484 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑠) = (𝐹𝑦))
8987, 88oveq12d 7433 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((𝐽𝑟)(+g𝐻)(𝐽𝑠)) = ((𝐹𝑥)(+g𝐻)(𝐹𝑦)))
9086, 89eqtr4d 2771 . . . . 5 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
916ad4antr 731 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
92 simpllr 775 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑠 ∈ (Base‘𝑄))
937, 91, 5, 11, 20, 92ghmquskerlem2 19230 . . . . 5 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ∃𝑦𝑠 (𝐽𝑠) = (𝐹𝑦))
9490, 93r19.29a 3158 . . . 4 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
9551adantr 480 . . . 4 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
9694, 95r19.29a 3158 . . 3 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
9796anasss 466 . 2 ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(+g𝑄)𝑠)) = ((𝐽𝑟)(+g𝐻)(𝐽𝑠)))
981, 2, 3, 4, 13, 16, 53, 97isghmd 19173 1 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3066  Vcvv 3470  wss 3945  𝒫 cpw 4599  {csn 4625   cuni 4904  cmpt 5226  ccnv 5672  ran crn 5674  cima 5676   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7415   Er wer 8716  [cec 8717   / cqs 8718  Basecbs 17174  +gcplusg 17227  0gc0g 17415   /s cqus 17481  Grpcgrp 18884  SubGrpcsubg 19069  NrmSGrpcnsg 19070   ~QG cqg 19071   GrpHom cghm 19161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7866  df-1st 7988  df-2nd 7989  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-1o 8481  df-er 8719  df-ec 8721  df-qs 8725  df-en 8959  df-dom 8960  df-sdom 8961  df-fin 8962  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17417  df-imas 17484  df-qus 17485  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-submnd 18735  df-grp 18887  df-minusg 18888  df-sbg 18889  df-subg 19072  df-nsg 19073  df-eqg 19074  df-ghm 19162
This theorem is referenced by:  ghmqusker  19232  rhmquskerlem  33135
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