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Theorem rhmimaidl 33460
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 20485 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4 fimass 6756 . . . . 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 6737 . . . . . 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 20476 . . . . . . 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 20264 . . . . . 6 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
1310, 12syl 17 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (0g𝑅) ∈ (Base‘𝑅))
14 simpr 484 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐼𝑇)
15 rhmimaidl.t . . . . . . 7 𝑇 = (LIdeal‘𝑅)
1615, 11lidl0cl 21230 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
1710, 14, 16syl2anc 584 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
188, 13, 17fnfvimad 7254 . . . 4 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹‘(0g𝑅)) ∈ (𝐹𝐼))
1918ne0d 4342 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ≠ ∅)
20 rhmghm 20484 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑧 ∈ (Base‘𝑅))
241, 15lidlss 21222 . . . . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28 eqid 2737 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
291, 28ringcl 20247 . . . . . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . . . . . 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 19239 . . . . . . . . . . . . . . . . . . 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 20486 . . . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 7449 . . . . . . . . . . . 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 7449 . . . . . . . . . . 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 21225 . . . . . . . . . . . . . . . . 17 (𝐼𝑇 ↔ (𝐼 ⊆ (Base‘𝑅) ∧ 𝐼 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼))
6362simp3bi 1148 . . . . . . . . . . . . . . . 16 (𝐼𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6463r19.21bi 3251 . . . . . . . . . . . . . . 15 ((𝐼𝑇𝑧 ∈ (Base‘𝑅)) → ∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6564r19.21bi 3251 . . . . . . . . . . . . . 14 (((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) → ∀𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6665r19.21bi 3251 . . . . . . . . . . . . 13 ((((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6758, 59, 60, 61, 66syl1111anc 841 . . . . . . . . . . . 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 3984 . . . . . . . . . . 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 7254 . . . . . . . . . 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 2842 . . . . . . . . 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 6740 . . . . . . . . . . 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 6746 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
7574imaeq2d 6078 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76 imadmrn 6088 . . . . . . . . . . . . . . . . 17 (𝐹 “ dom 𝐹) = ran 𝐹
7775, 76eqtr3di 2792 . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . 14 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝐹 “ (Base‘𝑅)) = 𝐵)
8079eleq2d 2827 . . . . . . . . . . . . 13 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) → (𝑥 ∈ (𝐹 “ (Base‘𝑅)) ↔ 𝑥𝐵))
8180biimpar 477 . . . . . . . . . . . 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 6974 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ (𝐹 “ (Base‘𝑅))) → ∃𝑧 ∈ (Base‘𝑅)(𝐹𝑧) = 𝑥)
8573, 83, 84syl2anc 584 . . . . . . . . 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 3162 . . . . . . . 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 6974 . . . . . . . . 9 ((Fun 𝐹𝑎 ∈ (𝐹𝐼)) → ∃𝑖𝐼 (𝐹𝑖) = 𝑎)
9087, 88, 89syl2anc 584 . . . . . . . 8 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → ∃𝑖𝐼 (𝐹𝑖) = 𝑎)
9186, 90r19.29a 3162 . . . . . . 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 484 . . . . . . . 8 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → 𝑏 ∈ (𝐹𝐼))
94 fvelima 6974 . . . . . . . 8 ((Fun 𝐹𝑏 ∈ (𝐹𝐼)) → ∃𝑗𝐼 (𝐹𝑗) = 𝑏)
9592, 93, 94syl2anc 584 . . . . . . 7 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → ∃𝑗𝐼 (𝐹𝑗) = 𝑏)
9691, 95r19.29a 3162 . . . . . 6 ((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9796anasss 466 . . . . 5 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ (𝑎 ∈ (𝐹𝐼) ∧ 𝑏 ∈ (𝐹𝐼))) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9897ralrimivva 3202 . . . 4 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → ∀𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9998ralrimiva 3146 . . 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 21225 . . 3 ((𝐹𝐼) ∈ 𝑈 ↔ ((𝐹𝐼) ⊆ 𝐵 ∧ (𝐹𝐼) ≠ ∅ ∧ ∀𝑥𝐵𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼)))
1026, 19, 99, 101syl3anbrc 1344 . 2 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
1031023impa 1110 1 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333  dom cdm 5685  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 17484   GrpHom cghm 19230  Ringcrg 20230   RingHom crh 20469  LIdealclidl 21216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-mgp 20138  df-ur 20179  df-ring 20232  df-rhm 20472  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218
This theorem is referenced by:  rhmpreimacnlem  33883  rhmpreimacn  33884
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