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Theorem rhmimaidl 33440
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 2735 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
2 rhmimaidl.b . . . . . 6 𝐵 = (Base‘𝑆)
31, 2rhmf 20502 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4 fimass 6757 . . . . 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 6738 . . . . . 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 20493 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
109ad2antrr 726 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝑅 ∈ Ring)
11 eqid 2735 . . . . . . 7 (0g𝑅) = (0g𝑅)
121, 11ring0cl 20281 . . . . . 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 21248 . . . . . 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 4348 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ≠ ∅)
20 rhmghm 20501 . . . . . . . . . . . . . . . . . . . 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 21240 . . . . . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28 eqid 2735 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
291, 28ringcl 20268 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅))
3022, 23, 27, 29syl3anc 1370 . . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (+g𝑅) = (+g𝑅)
34 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (+g𝑆) = (+g𝑆)
351, 33, 34ghmlin 19252 . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑆) = (.r𝑆)
391, 28, 38rhmmul 20503 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(.r𝑅)𝑖)) = ((𝐹𝑧)(.r𝑆)(𝐹𝑖)))
4037, 23, 27, 39syl3anc 1370 . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . 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 21243 . . . . . . . . . . . . . . . . 17 (𝐼𝑇 ↔ (𝐼 ⊆ (Base‘𝑅) ∧ 𝐼 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼))
6362simp3bi 1146 . . . . . . . . . . . . . . . 16 (𝐼𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6463r19.21bi 3249 . . . . . . . . . . . . . . 15 ((𝐼𝑇𝑧 ∈ (Base‘𝑅)) → ∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6564r19.21bi 3249 . . . . . . . . . . . . . 14 (((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) → ∀𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6665r19.21bi 3249 . . . . . . . . . . . . 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 3996 . . . . . . . . . . 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 2840 . . . . . . . . 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 6741 . . . . . . . . . . 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 6747 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
7574imaeq2d 6080 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
76 imadmrn 6090 . . . . . . . . . . . . . . . . 17 (𝐹 “ dom 𝐹) = ran 𝐹
7775, 76eqtr3di 2790 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
7877eqeq1d 2737 . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . 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 3160 . . . . . . . 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 3160 . . . . . . 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 3160 . . . . . 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 3200 . . . 4 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → ∀𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9998ralrimiva 3144 . . 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 21243 . . 3 ((𝐹𝐼) ∈ 𝑈 ↔ ((𝐹𝐼) ⊆ 𝐵 ∧ (𝐹𝐼) ≠ ∅ ∧ ∀𝑥𝐵𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼)))
1026, 19, 99, 101syl3anbrc 1342 . 2 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
1031023impa 1109 1 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  .rcmulr 17299  0gc0g 17486   GrpHom cghm 19243  Ringcrg 20251   RingHom crh 20486  LIdealclidl 21234
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-op 4638  df-uni 4913  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-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  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-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-0g 17488  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-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-rhm 20489  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236
This theorem is referenced by:  rhmpreimacnlem  33845  rhmpreimacn  33846
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