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Theorem rhmquskerlem 33140
Description: The mapping 𝐽 induced by a ring homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 22-Mar-2025.)
Hypotheses
Ref Expression
rhmqusker.1 0 = (0g𝐻)
rhmqusker.f (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusker.k 𝐾 = (𝐹 “ { 0 })
rhmqusker.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
rhmquskerlem.j 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
rhmquskerlem.2 (𝜑𝐺 ∈ CRing)
Assertion
Ref Expression
rhmquskerlem (𝜑𝐽 ∈ (𝑄 RingHom 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 rhmquskerlem
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 (1r𝑄) = (1r𝑄)
3 eqid 2728 . 2 (1r𝐻) = (1r𝐻)
4 eqid 2728 . 2 (.r𝑄) = (.r𝑄)
5 eqid 2728 . 2 (.r𝐻) = (.r𝐻)
6 rhmqusker.f . . . . 5 (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
7 rhmrcl1 20414 . . . . 5 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
86, 7syl 17 . . . 4 (𝜑𝐺 ∈ Ring)
9 rhmqusker.k . . . . . 6 𝐾 = (𝐹 “ { 0 })
10 eqid 2728 . . . . . . . 8 (LIdeal‘𝐺) = (LIdeal‘𝐺)
11 rhmqusker.1 . . . . . . . 8 0 = (0g𝐻)
1210, 11kerlidl 33134 . . . . . . 7 (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
136, 12syl 17 . . . . . 6 (𝜑 → (𝐹 “ { 0 }) ∈ (LIdeal‘𝐺))
149, 13eqeltrid 2833 . . . . 5 (𝜑𝐾 ∈ (LIdeal‘𝐺))
15 rhmquskerlem.2 . . . . . 6 (𝜑𝐺 ∈ CRing)
1610crng2idl 21172 . . . . . 6 (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
1715, 16syl 17 . . . . 5 (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
1814, 17eleqtrd 2831 . . . 4 (𝜑𝐾 ∈ (2Ideal‘𝐺))
19 rhmqusker.q . . . . 5 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
20 eqid 2728 . . . . 5 (2Ideal‘𝐺) = (2Ideal‘𝐺)
21 eqid 2728 . . . . 5 (1r𝐺) = (1r𝐺)
2219, 20, 21qus1 21167 . . . 4 ((𝐺 ∈ Ring ∧ 𝐾 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝐾) = (1r𝑄)))
238, 18, 22syl2anc 583 . . 3 (𝜑 → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝐾) = (1r𝑄)))
2423simpld 494 . 2 (𝜑𝑄 ∈ Ring)
25 rhmrcl2 20415 . . 3 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
266, 25syl 17 . 2 (𝜑𝐻 ∈ Ring)
27 rhmghm 20422 . . . . 5 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
286, 27syl 17 . . . 4 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
29 rhmquskerlem.j . . . 4 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
30 eqid 2728 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3130, 21ringidcl 20201 . . . . 5 (𝐺 ∈ Ring → (1r𝐺) ∈ (Base‘𝐺))
328, 31syl 17 . . . 4 (𝜑 → (1r𝐺) ∈ (Base‘𝐺))
3311, 28, 9, 19, 29, 32ghmquskerlem1 19233 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝐾)) = (𝐹‘(1r𝐺)))
3423simprd 495 . . . 4 (𝜑 → [(1r𝐺)](𝐺 ~QG 𝐾) = (1r𝑄))
3534fveq2d 6901 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝐾)) = (𝐽‘(1r𝑄)))
3621, 3rhm1 20427 . . . 4 (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r𝐺)) = (1r𝐻))
376, 36syl 17 . . 3 (𝜑 → (𝐹‘(1r𝐺)) = (1r𝐻))
3833, 35, 373eqtr3d 2776 . 2 (𝜑 → (𝐽‘(1r𝑄)) = (1r𝐻))
396ad6antr 735 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
4019a1i 11 . . . . . . . . . . . . 13 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
41 eqidd 2729 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42 ovexd 7455 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) ∈ V)
4340, 41, 42, 15qusbas 17526 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
4411ghmker 19195 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
4528, 44syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹 “ { 0 }) ∈ (NrmSGrp‘𝐺))
469, 45eqeltrid 2833 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ (NrmSGrp‘𝐺))
47 nsgsubg 19112 . . . . . . . . . . . . . 14 (𝐾 ∈ (NrmSGrp‘𝐺) → 𝐾 ∈ (SubGrp‘𝐺))
48 eqid 2728 . . . . . . . . . . . . . . 15 (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
4930, 48eqger 19132 . . . . . . . . . . . . . 14 (𝐾 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
5046, 47, 493syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
5150qsss 8796 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ⊆ 𝒫 (Base‘𝐺))
5243, 51eqsstrrd 4019 . . . . . . . . . . 11 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
5352sselda 3980 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
5453elpwid 4612 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
5554ad5antr 733 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ⊆ (Base‘𝐺))
56 simp-4r 783 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥𝑟)
5755, 56sseldd 3981 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥 ∈ (Base‘𝐺))
5852sselda 3980 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
5958elpwid 4612 . . . . . . . . . 10 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
6059adantlr 714 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
6160ad4antr 731 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ⊆ (Base‘𝐺))
62 simplr 768 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦𝑠)
6361, 62sseldd 3981 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦 ∈ (Base‘𝐺))
64 eqid 2728 . . . . . . . 8 (.r𝐺) = (.r𝐺)
6530, 64, 5rhmmul 20424 . . . . . . 7 ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6639, 57, 63, 65syl3anc 1369 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6750ad6antr 735 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐺 ~QG 𝐾) Er (Base‘𝐺))
68 simp-6r 787 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ (Base‘𝑄))
6943ad6antr 735 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝐾)) = (Base‘𝑄))
7068, 69eleqtrrd 2832 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
71 qsel 8814 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
7267, 70, 56, 71syl3anc 1369 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝐾))
73 simp-5r 785 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ (Base‘𝑄))
7473, 69eleqtrrd 2832 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)))
75 qsel 8814 . . . . . . . . . . 11 (((𝐺 ~QG 𝐾) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝐾)) ∧ 𝑦𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
7667, 74, 62, 75syl3anc 1369 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝐾))
7772, 76oveq12d 7438 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(.r𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝐾)(.r𝑄)[𝑦](𝐺 ~QG 𝐾)))
7815ad6antr 735 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ CRing)
7914ad6antr 735 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐾 ∈ (LIdeal‘𝐺))
8019, 30, 64, 4, 78, 79, 57, 63qusmul 33114 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ([𝑥](𝐺 ~QG 𝐾)(.r𝑄)[𝑦](𝐺 ~QG 𝐾)) = [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝐾))
8177, 80eqtr2d 2769 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝐾) = (𝑟(.r𝑄)𝑠))
8281fveq2d 6901 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝐾)) = (𝐽‘(𝑟(.r𝑄)𝑠)))
8339, 27syl 17 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8439, 7syl 17 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ Ring)
8530, 64, 84, 57, 63ringcld 20198 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑥(.r𝐺)𝑦) ∈ (Base‘𝐺))
8611, 83, 9, 19, 29, 85ghmquskerlem1 19233 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝐾)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
8782, 86eqtr3d 2770 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
88 simpllr 775 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑟) = (𝐹𝑥))
89 simpr 484 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽𝑠) = (𝐹𝑦))
9088, 89oveq12d 7438 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((𝐽𝑟)(.r𝐻)(𝐽𝑠)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
9166, 87, 903eqtr4d 2778 . . . . 5 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9228ad4antr 731 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
93 simpllr 775 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑠 ∈ (Base‘𝑄))
9411, 92, 9, 19, 29, 93ghmquskerlem2 19235 . . . . 5 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ∃𝑦𝑠 (𝐽𝑠) = (𝐹𝑦))
9591, 94r19.29a 3159 . . . 4 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9628ad2antrr 725 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
97 simplr 768 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
9811, 96, 9, 19, 29, 97ghmquskerlem2 19235 . . . 4 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
9995, 98r19.29a 3159 . . 3 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
10099anasss 466 . 2 ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
10111, 28, 9, 19, 29ghmquskerlem3 19236 . 2 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
1021, 2, 3, 4, 5, 24, 26, 38, 100, 101isrhm2d 20425 1 (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  wss 3947  𝒫 cpw 4603  {csn 4629   cuni 4908  cmpt 5231  ccnv 5677  cima 5681  cfv 6548  (class class class)co 7420   Er wer 8721  [cec 8722   / cqs 8723  Basecbs 17179  .rcmulr 17233  0gc0g 17420   /s cqus 17486  SubGrpcsubg 19074  NrmSGrpcnsg 19075   ~QG cqg 19076   GrpHom cghm 19166  1rcur 20120  Ringcrg 20172  CRingccrg 20173   RingHom crh 20407  LIdealclidl 21101  2Idealc2idl 21142
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
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 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-tpos 8231  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-ec 8726  df-qs 8730  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9465  df-inf 9466  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-0g 17422  df-imas 17489  df-qus 17490  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18739  df-submnd 18740  df-grp 18892  df-minusg 18893  df-sbg 18894  df-subg 19077  df-nsg 19078  df-eqg 19079  df-ghm 19167  df-cmn 19736  df-abl 19737  df-mgp 20074  df-rng 20092  df-ur 20121  df-ring 20174  df-cring 20175  df-oppr 20272  df-rhm 20410  df-subrg 20507  df-lmod 20744  df-lss 20815  df-lsp 20855  df-sra 21057  df-rgmod 21058  df-lidl 21103  df-rsp 21104  df-2idl 21143
This theorem is referenced by:  rhmqusker  33141  algextdeglem4  33388
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