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Theorem rhmimaidl 31323
Description: The image of an ideal 𝐼 by a surjective ring homomorphism 𝐹 is an ideal. (Contributed by Thierry Arnoux, 6-Jul-2024.)
Hypotheses
Ref Expression
rhmimaidl.b 𝐵 = (Base‘𝑆)
rhmimaidl.t 𝑇 = (LIdeal‘𝑅)
rhmimaidl.u 𝑈 = (LIdeal‘𝑆)
Assertion
Ref Expression
rhmimaidl ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)

Proof of Theorem rhmimaidl
Dummy variables 𝑎 𝑏 𝑖 𝑗 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
2 rhmimaidl.b . . . . . 6 𝐵 = (Base‘𝑆)
31, 2rhmf 19746 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4 fimass 6566 . . . . 5 (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹𝐼) ⊆ 𝐵)
53, 4syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹𝐼) ⊆ 𝐵)
65ad2antrr 726 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ⊆ 𝐵)
73ffnd 6546 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
87ad2antrr 726 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐹 Fn (Base‘𝑅))
9 rhmrcl1 19739 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
109ad2antrr 726 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝑅 ∈ Ring)
11 eqid 2737 . . . . . . 7 (0g𝑅) = (0g𝑅)
121, 11ring0cl 19587 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1310, 12syl 17 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (0g𝑅) ∈ (Base‘𝑅))
14 simpr 488 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐼𝑇)
15 rhmimaidl.t . . . . . . 7 𝑇 = (LIdeal‘𝑅)
1615, 11lidl0cl 20250 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
1710, 14, 16syl2anc 587 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
188, 13, 17fnfvimad 7050 . . . 4 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹‘(0g𝑅)) ∈ (𝐹𝐼))
1918ne0d 4250 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ≠ ∅)
20 rhmghm 19745 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2120ad4antr 732 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
229ad4antr 732 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
23 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
241, 15lidlss 20248 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝑇𝐼 ⊆ (Base‘𝑅))
2524ad4antlr 733 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26 simplr 769 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖𝐼)
2725, 26sseldd 3902 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
291, 28ringcl 19579 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅))
3022, 23, 27, 29syl3anc 1373 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅))
31 simpllr 776 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗𝐼)
3225, 31sseldd 3902 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (+g𝑅) = (+g𝑅)
34 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (+g𝑆) = (+g𝑆)
351, 33, 34ghmlin 18627 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅) ∧ 𝑗 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = ((𝐹‘(𝑧(.r𝑅)𝑖))(+g𝑆)(𝐹𝑗)))
3621, 30, 32, 35syl3anc 1373 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = ((𝐹‘(𝑧(.r𝑅)𝑖))(+g𝑆)(𝐹𝑗)))
37 simp-4l 783 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑆) = (.r𝑆)
391, 28, 38rhmmul 19747 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(.r𝑅)𝑖)) = ((𝐹𝑧)(.r𝑆)(𝐹𝑖)))
4037, 23, 27, 39syl3anc 1373 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(.r𝑅)𝑖)) = ((𝐹𝑧)(.r𝑆)(𝐹𝑖)))
4140oveq1d 7228 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹‘(𝑧(.r𝑅)𝑖))(+g𝑆)(𝐹𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4236, 41eqtrd 2777 . . . . . . . . . . . . . . . . 17 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4342adantl4r 755 . . . . . . . . . . . . . . . 16 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4443adantl3r 750 . . . . . . . . . . . . . . 15 (((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4544adantl3r 750 . . . . . . . . . . . . . 14 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4645adantl3r 750 . . . . . . . . . . . . 13 (((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4746adantllr 719 . . . . . . . . . . . 12 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4847ad4ant13 751 . . . . . . . . . . 11 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
49 simpr 488 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹𝑧) = 𝑥)
50 simpllr 776 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹𝑖) = 𝑎)
5149, 50oveq12d 7231 . . . . . . . . . . . 12 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → ((𝐹𝑧)(.r𝑆)(𝐹𝑖)) = (𝑥(.r𝑆)𝑎))
52 simp-5r 786 . . . . . . . . . . . 12 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹𝑗) = 𝑏)
5351, 52oveq12d 7231 . . . . . . . . . . 11 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
5448, 53eqtrd 2777 . . . . . . . . . 10 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏))
558ad9antr 742 . . . . . . . . . . 11 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝐹 Fn (Base‘𝑅))
5614, 24syl 17 . . . . . . . . . . . . 13 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐼 ⊆ (Base‘𝑅))
5756ad9antr 742 . . . . . . . . . . . 12 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝐼 ⊆ (Base‘𝑅))
5814ad9antr 742 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝐼𝑇)
59 simplr 769 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝑧 ∈ (Base‘𝑅))
60 simp-4r 784 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝑖𝐼)
61 simp-6r 788 . . . . . . . . . . . . 13 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → 𝑗𝐼)
6215, 1, 33, 28islidl 20249 . . . . . . . . . . . . . . . . 17 (𝐼𝑇 ↔ (𝐼 ⊆ (Base‘𝑅) ∧ 𝐼 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼))
6362simp3bi 1149 . . . . . . . . . . . . . . . 16 (𝐼𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6463r19.21bi 3130 . . . . . . . . . . . . . . 15 ((𝐼𝑇𝑧 ∈ (Base‘𝑅)) → ∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6564r19.21bi 3130 . . . . . . . . . . . . . 14 (((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) → ∀𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6665r19.21bi 3130 . . . . . . . . . . . . 13 ((((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6758, 59, 60, 61, 66syl1111anc 840 . . . . . . . . . . . 12 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6857, 67sseldd 3902 . . . . . . . . . . 11 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ (Base‘𝑅))
6955, 68, 67fnfvimad 7050 . . . . . . . . . 10 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) ∈ (𝐹𝐼))
7054, 69eqeltrrd 2839 . . . . . . . . 9 ((((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑧) = 𝑥) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
713ad2antrr 726 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
7271ffund 6549 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → Fun 𝐹)
7372ad7antr 738 . . . . . . . . . 10 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) → Fun 𝐹)
743fdmd 6556 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
7574imaeq2d 5929 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76 imadmrn 5939 . . . . . . . . . . . . . . . . 17 (𝐹 “ dom 𝐹) = ran 𝐹
7775, 76eqtr3di 2793 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
7877eqeq1d 2739 . . . . . . . . . . . . . . 15 (𝐹 ∈ (𝑅 RingHom 𝑆) → ((𝐹 “ (Base‘𝑅)) = 𝐵 ↔ ran 𝐹 = 𝐵))
7978biimpar 481 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
8079eleq2d 2823 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥𝐵))
8180biimpar 481 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
8281adantlr 715 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
8382ad6antr 736 . . . . . . . . . 10 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
84 fvelima 6778 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹𝑧) = 𝑥)
8573, 83, 84syl2anc 587 . . . . . . . . 9 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) → ∃𝑧 ∈ (Base‘𝑅)(𝐹𝑧) = 𝑥)
8670, 85r19.29a 3208 . . . . . . . 8 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
8772ad5antr 734 . . . . . . . . 9 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → Fun 𝐹)
88 simp-4r 784 . . . . . . . . 9 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → 𝑎 ∈ (𝐹𝐼))
89 fvelima 6778 . . . . . . . . 9 ((Fun 𝐹𝑎 ∈ (𝐹𝐼)) → ∃𝑖𝐼 (𝐹𝑖) = 𝑎)
9087, 88, 89syl2anc 587 . . . . . . . 8 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → ∃𝑖𝐼 (𝐹𝑖) = 𝑎)
9186, 90r19.29a 3208 . . . . . . 7 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9272ad3antrrr 730 . . . . . . . 8 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → Fun 𝐹)
93 simpr 488 . . . . . . . 8 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → 𝑏 ∈ (𝐹𝐼))
94 fvelima 6778 . . . . . . . 8 ((Fun 𝐹𝑏 ∈ (𝐹𝐼)) → ∃𝑗𝐼 (𝐹𝑗) = 𝑏)
9592, 93, 94syl2anc 587 . . . . . . 7 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → ∃𝑗𝐼 (𝐹𝑗) = 𝑏)
9691, 95r19.29a 3208 . . . . . 6 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9796anasss 470 . . . . 5 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ (𝑎 ∈ (𝐹𝐼) ∧ 𝑏 ∈ (𝐹𝐼))) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9897ralrimivva 3112 . . . 4 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → ∀𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9998ralrimiva 3105 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → ∀𝑥𝐵𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
100 rhmimaidl.u . . . 4 𝑈 = (LIdeal‘𝑆)
101100, 2, 34, 38islidl 20249 . . 3 ((𝐹𝐼) ∈ 𝑈 ↔ ((𝐹𝐼) ⊆ 𝐵 ∧ (𝐹𝐼) ≠ ∅ ∧ ∀𝑥𝐵𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼)))
1026, 19, 99, 101syl3anbrc 1345 . 2 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
1031023impa 1112 1 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  wral 3061  wrex 3062  wss 3866  c0 4237  dom cdm 5551  ran crn 5552  cima 5554  Fun wfun 6374   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  +gcplusg 16802  .rcmulr 16803  0gc0g 16944   GrpHom cghm 18619  Ringcrg 19562   RingHom crh 19732  LIdealclidl 20207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-sca 16818  df-vsca 16819  df-ip 16820  df-0g 16946  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-grp 18368  df-minusg 18369  df-sbg 18370  df-subg 18540  df-ghm 18620  df-mgp 19505  df-ur 19517  df-ring 19564  df-rnghom 19735  df-subrg 19798  df-lmod 19901  df-lss 19969  df-sra 20209  df-rgmod 20210  df-lidl 20211
This theorem is referenced by:  rhmpreimacnlem  31548  rhmpreimacn  31549
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