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Theorem rhmqusnsg 21313
Description: The mapping 𝐽 induced by a ring homomorphism 𝐹 from a subring 𝑁 of the quotient group 𝑄 over 𝐹's kernel 𝐾 is a ring homomorphism. (Contributed by Thierry Arnoux, 13-May-2025.)
Hypotheses
Ref Expression
rhmqusnsg.0 0 = (0g𝐻)
rhmqusnsg.f (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
rhmqusnsg.k 𝐾 = (𝐹 “ { 0 })
rhmqusnsg.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
rhmqusnsg.j 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
rhmqusnsg.g (𝜑𝐺 ∈ CRing)
rhmqusnsg.n (𝜑𝑁𝐾)
rhmqusnsg.1 (𝜑𝑁 ∈ (LIdeal‘𝐺))
Assertion
Ref Expression
rhmqusnsg (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
Distinct variable groups:   𝐹,𝑞   𝐺,𝑞   𝐻,𝑞   𝐽,𝑞   𝐾,𝑞   𝑁,𝑞   𝑄,𝑞   𝜑,𝑞
Allowed substitution hint:   0 (𝑞)

Proof of Theorem rhmqusnsg
Dummy variables 𝑟 𝑥 𝑦 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . 2 (Base‘𝑄) = (Base‘𝑄)
2 eqid 2735 . 2 (1r𝑄) = (1r𝑄)
3 eqid 2735 . 2 (1r𝐻) = (1r𝐻)
4 eqid 2735 . 2 (.r𝑄) = (.r𝑄)
5 eqid 2735 . 2 (.r𝐻) = (.r𝐻)
6 rhmqusnsg.g . . . . 5 (𝜑𝐺 ∈ CRing)
76crngringd 20264 . . . 4 (𝜑𝐺 ∈ Ring)
8 rhmqusnsg.1 . . . . 5 (𝜑𝑁 ∈ (LIdeal‘𝐺))
9 eqid 2735 . . . . . . 7 (LIdeal‘𝐺) = (LIdeal‘𝐺)
109crng2idl 21309 . . . . . 6 (𝐺 ∈ CRing → (LIdeal‘𝐺) = (2Ideal‘𝐺))
116, 10syl 17 . . . . 5 (𝜑 → (LIdeal‘𝐺) = (2Ideal‘𝐺))
128, 11eleqtrd 2841 . . . 4 (𝜑𝑁 ∈ (2Ideal‘𝐺))
13 rhmqusnsg.q . . . . 5 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
14 eqid 2735 . . . . 5 (2Ideal‘𝐺) = (2Ideal‘𝐺)
15 eqid 2735 . . . . 5 (1r𝐺) = (1r𝐺)
1613, 14, 15qus1 21302 . . . 4 ((𝐺 ∈ Ring ∧ 𝑁 ∈ (2Ideal‘𝐺)) → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄)))
177, 12, 16syl2anc 584 . . 3 (𝜑 → (𝑄 ∈ Ring ∧ [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄)))
1817simpld 494 . 2 (𝜑𝑄 ∈ Ring)
19 rhmqusnsg.f . . 3 (𝜑𝐹 ∈ (𝐺 RingHom 𝐻))
20 rhmrcl2 20494 . . 3 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐻 ∈ Ring)
2119, 20syl 17 . 2 (𝜑𝐻 ∈ Ring)
22 rhmqusnsg.0 . . . 4 0 = (0g𝐻)
23 rhmghm 20501 . . . . 5 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
2419, 23syl 17 . . . 4 (𝜑𝐹 ∈ (𝐺 GrpHom 𝐻))
25 rhmqusnsg.k . . . 4 𝐾 = (𝐹 “ { 0 })
26 rhmqusnsg.j . . . 4 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ (𝐹𝑞))
27 rhmqusnsg.n . . . 4 (𝜑𝑁𝐾)
28 lidlnsg 21276 . . . . 5 ((𝐺 ∈ Ring ∧ 𝑁 ∈ (LIdeal‘𝐺)) → 𝑁 ∈ (NrmSGrp‘𝐺))
297, 8, 28syl2anc 584 . . . 4 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
30 eqid 2735 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
3130, 15ringidcl 20280 . . . . 5 (𝐺 ∈ Ring → (1r𝐺) ∈ (Base‘𝐺))
327, 31syl 17 . . . 4 (𝜑 → (1r𝐺) ∈ (Base‘𝐺))
3322, 24, 25, 13, 26, 27, 29, 32ghmqusnsglem1 19311 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝑁)) = (𝐹‘(1r𝐺)))
3417simprd 495 . . . 4 (𝜑 → [(1r𝐺)](𝐺 ~QG 𝑁) = (1r𝑄))
3534fveq2d 6911 . . 3 (𝜑 → (𝐽‘[(1r𝐺)](𝐺 ~QG 𝑁)) = (𝐽‘(1r𝑄)))
3615, 3rhm1 20506 . . . 4 (𝐹 ∈ (𝐺 RingHom 𝐻) → (𝐹‘(1r𝐺)) = (1r𝐻))
3719, 36syl 17 . . 3 (𝜑 → (𝐹‘(1r𝐺)) = (1r𝐻))
3833, 35, 373eqtr3d 2783 . 2 (𝜑 → (𝐽‘(1r𝑄)) = (1r𝐻))
3919ad6antr 736 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 RingHom 𝐻))
4013a1i 11 . . . . . . . . . . . . 13 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
41 eqidd 2736 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐺) = (Base‘𝐺))
42 ovexd 7466 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
4340, 41, 42, 6qusbas 17592 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
44 nsgsubg 19189 . . . . . . . . . . . . . 14 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
45 eqid 2735 . . . . . . . . . . . . . . 15 (𝐺 ~QG 𝑁) = (𝐺 ~QG 𝑁)
4630, 45eqger 19209 . . . . . . . . . . . . . 14 (𝑁 ∈ (SubGrp‘𝐺) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
4729, 44, 463syl 18 . . . . . . . . . . . . 13 (𝜑 → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
4847qsss 8817 . . . . . . . . . . . 12 (𝜑 → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ⊆ 𝒫 (Base‘𝐺))
4943, 48eqsstrrd 4035 . . . . . . . . . . 11 (𝜑 → (Base‘𝑄) ⊆ 𝒫 (Base‘𝐺))
5049sselda 3995 . . . . . . . . . 10 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ∈ 𝒫 (Base‘𝐺))
5150elpwid 4614 . . . . . . . . 9 ((𝜑𝑟 ∈ (Base‘𝑄)) → 𝑟 ⊆ (Base‘𝐺))
5251ad5antr 734 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ⊆ (Base‘𝐺))
53 simp-4r 784 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥𝑟)
5452, 53sseldd 3996 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑥 ∈ (Base‘𝐺))
5549sselda 3995 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ∈ 𝒫 (Base‘𝐺))
5655elpwid 4614 . . . . . . . . . 10 ((𝜑𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
5756adantlr 715 . . . . . . . . 9 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑠 ⊆ (Base‘𝐺))
5857ad4antr 732 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ⊆ (Base‘𝐺))
59 simplr 769 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦𝑠)
6058, 59sseldd 3996 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑦 ∈ (Base‘𝐺))
61 eqid 2735 . . . . . . . 8 (.r𝐺) = (.r𝐺)
6230, 61, 5rhmmul 20503 . . . . . . 7 ((𝐹 ∈ (𝐺 RingHom 𝐻) ∧ 𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6339, 54, 60, 62syl3anc 1370 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐹‘(𝑥(.r𝐺)𝑦)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
6447ad6antr 736 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐺 ~QG 𝑁) Er (Base‘𝐺))
65 simp-6r 788 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ (Base‘𝑄))
6643ad6antr 736 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((Base‘𝐺) / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
6765, 66eleqtrrd 2842 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
68 qsel 8835 . . . . . . . . . . 11 (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑟 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑥𝑟) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
6964, 67, 53, 68syl3anc 1370 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑟 = [𝑥](𝐺 ~QG 𝑁))
70 simp-5r 786 . . . . . . . . . . . 12 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ (Base‘𝑄))
7170, 66eleqtrrd 2842 . . . . . . . . . . 11 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)))
72 qsel 8835 . . . . . . . . . . 11 (((𝐺 ~QG 𝑁) Er (Base‘𝐺) ∧ 𝑠 ∈ ((Base‘𝐺) / (𝐺 ~QG 𝑁)) ∧ 𝑦𝑠) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
7364, 71, 59, 72syl3anc 1370 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑠 = [𝑦](𝐺 ~QG 𝑁))
7469, 73oveq12d 7449 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑟(.r𝑄)𝑠) = ([𝑥](𝐺 ~QG 𝑁)(.r𝑄)[𝑦](𝐺 ~QG 𝑁)))
756ad6antr 736 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ CRing)
768ad6antr 736 . . . . . . . . . 10 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁 ∈ (LIdeal‘𝐺))
7713, 30, 61, 4, 75, 76, 54, 60qusmulcrng 21312 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ([𝑥](𝐺 ~QG 𝑁)(.r𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁))
7874, 77eqtr2d 2776 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → [(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁) = (𝑟(.r𝑄)𝑠))
7978fveq2d 6911 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐽‘(𝑟(.r𝑄)𝑠)))
8039, 23syl 17 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
8127ad6antr 736 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁𝐾)
8229ad6antr 736 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝑁 ∈ (NrmSGrp‘𝐺))
83 rhmrcl1 20493 . . . . . . . . . 10 (𝐹 ∈ (𝐺 RingHom 𝐻) → 𝐺 ∈ Ring)
8439, 83syl 17 . . . . . . . . 9 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → 𝐺 ∈ Ring)
8530, 61, 84, 54, 60ringcld 20277 . . . . . . . 8 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝑥(.r𝐺)𝑦) ∈ (Base‘𝐺))
8622, 80, 25, 13, 26, 81, 82, 85ghmqusnsglem1 19311 . . . . . . 7 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘[(𝑥(.r𝐺)𝑦)](𝐺 ~QG 𝑁)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
8779, 86eqtr3d 2777 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = (𝐹‘(𝑥(.r𝐺)𝑦)))
88 simpllr 776 . . . . . . 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 7449 . . . . . 6 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → ((𝐽𝑟)(.r𝐻)(𝐽𝑠)) = ((𝐹𝑥)(.r𝐻)(𝐹𝑦)))
9163, 87, 903eqtr4d 2785 . . . . 5 (((((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) ∧ 𝑦𝑠) ∧ (𝐽𝑠) = (𝐹𝑦)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9224ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
9327ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑁𝐾)
9429ad4antr 732 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑁 ∈ (NrmSGrp‘𝐺))
95 simpllr 776 . . . . . 6 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → 𝑠 ∈ (Base‘𝑄))
9622, 92, 25, 13, 26, 93, 94, 95ghmqusnsglem2 19312 . . . . 5 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → ∃𝑦𝑠 (𝐽𝑠) = (𝐹𝑦))
9791, 96r19.29a 3160 . . . 4 (((((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) ∧ 𝑥𝑟) ∧ (𝐽𝑟) = (𝐹𝑥)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
9824ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
9927ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁𝐾)
10029ad2antrr 726 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑁 ∈ (NrmSGrp‘𝐺))
101 simplr 769 . . . . 5 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → 𝑟 ∈ (Base‘𝑄))
10222, 98, 25, 13, 26, 99, 100, 101ghmqusnsglem2 19312 . . . 4 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → ∃𝑥𝑟 (𝐽𝑟) = (𝐹𝑥))
10397, 102r19.29a 3160 . . 3 (((𝜑𝑟 ∈ (Base‘𝑄)) ∧ 𝑠 ∈ (Base‘𝑄)) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
104103anasss 466 . 2 ((𝜑 ∧ (𝑟 ∈ (Base‘𝑄) ∧ 𝑠 ∈ (Base‘𝑄))) → (𝐽‘(𝑟(.r𝑄)𝑠)) = ((𝐽𝑟)(.r𝐻)(𝐽𝑠)))
10522, 24, 25, 13, 26, 27, 29ghmqusnsg 19313 . 2 (𝜑𝐽 ∈ (𝑄 GrpHom 𝐻))
1061, 2, 3, 4, 5, 18, 21, 38, 104, 105isrhm2d 20504 1 (𝜑𝐽 ∈ (𝑄 RingHom 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  cmpt 5231  ccnv 5688  cima 5692  cfv 6563  (class class class)co 7431   Er wer 8741  [cec 8742   / cqs 8743  Basecbs 17245  .rcmulr 17299  0gc0g 17486   /s cqus 17552  SubGrpcsubg 19151  NrmSGrpcnsg 19152   ~QG cqg 19153   GrpHom cghm 19243  1rcur 20199  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486  LIdealclidl 21234  2Idealc2idl 21277
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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3045  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-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  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-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  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-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-rhm 20489  df-subrg 20587  df-lmod 20877  df-lss 20948  df-lsp 20988  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278
This theorem is referenced by:  zndvdchrrhm  41953  rhmqusspan  42167
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