Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmimaidl Structured version   Visualization version   GIF version

Theorem rhmimaidl 31075
 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 2798 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
2 rhmimaidl.b . . . . . 6 𝐵 = (Base‘𝑆)
31, 2rhmf 19492 . . . . 5 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:(Base‘𝑅)⟶𝐵)
4 fimass 6534 . . . . 5 (𝐹:(Base‘𝑅)⟶𝐵 → (𝐹𝐼) ⊆ 𝐵)
53, 4syl 17 . . . 4 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹𝐼) ⊆ 𝐵)
65ad2antrr 725 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ⊆ 𝐵)
73ffnd 6493 . . . . . 6 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 Fn (Base‘𝑅))
87ad2antrr 725 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐹 Fn (Base‘𝑅))
9 rhmrcl1 19485 . . . . . . 7 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
109ad2antrr 725 . . . . . 6 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝑅 ∈ Ring)
11 eqid 2798 . . . . . . 7 (0g𝑅) = (0g𝑅)
121, 11ring0cl 19333 . . . . . 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 19996 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
1710, 14, 16syl2anc 587 . . . . 5 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (0g𝑅) ∈ 𝐼)
188, 13, 17fnfvimad 6981 . . . 4 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹‘(0g𝑅)) ∈ (𝐹𝐼))
1918ne0d 4253 . . 3 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ≠ ∅)
20 rhmghm 19491 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
2120ad4antr 731 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 GrpHom 𝑆))
229ad4antr 731 . . . . . . . . . . . . . . . . . . . 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 19994 . . . . . . . . . . . . . . . . . . . . . 22 (𝐼𝑇𝐼 ⊆ (Base‘𝑅))
2524ad4antlr 732 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐼 ⊆ (Base‘𝑅))
26 simplr 768 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖𝐼)
2725, 26sseldd 3917 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑖 ∈ (Base‘𝑅))
28 eqid 2798 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑅) = (.r𝑅)
291, 28ringcl 19325 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ Ring ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅))
3022, 23, 27, 29syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝑧(.r𝑅)𝑖) ∈ (Base‘𝑅))
31 simpllr 775 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗𝐼)
3225, 31sseldd 3917 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝑗 ∈ (Base‘𝑅))
33 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (+g𝑅) = (+g𝑅)
34 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (+g𝑆) = (+g𝑆)
351, 33, 34ghmlin 18373 . . . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . . . . . . 20 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → 𝐹 ∈ (𝑅 RingHom 𝑆))
38 eqid 2798 . . . . . . . . . . . . . . . . . . . . 21 (.r𝑆) = (.r𝑆)
391, 28, 38rhmmul 19493 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝑧 ∈ (Base‘𝑅) ∧ 𝑖 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(.r𝑅)𝑖)) = ((𝐹𝑧)(.r𝑆)(𝐹𝑖)))
4037, 23, 27, 39syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘(𝑧(.r𝑅)𝑖)) = ((𝐹𝑧)(.r𝑆)(𝐹𝑖)))
4140oveq1d 7157 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹‘(𝑧(.r𝑅)𝑖))(+g𝑆)(𝐹𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4236, 41eqtrd 2833 . . . . . . . . . . . . . . . . 17 (((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐼𝑇) ∧ 𝑗𝐼) ∧ 𝑖𝐼) ∧ 𝑧 ∈ (Base‘𝑅)) → (𝐹‘((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗)) = (((𝐹𝑧)(.r𝑆)(𝐹𝑖))(+g𝑆)(𝐹𝑗)))
4342adantl4r 754 . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . 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 718 . . . . . . . . . . . 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 750 . . . . . . . . . . 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 775 . . . . . . . . . . . . 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 7160 . . . . . . . . . . . 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 785 . . . . . . . . . . . 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 7160 . . . . . . . . . . 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 2833 . . . . . . . . . 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 741 . . . . . . . . . . 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 741 . . . . . . . . . . . 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 741 . . . . . . . . . . . . 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 768 . . . . . . . . . . . . 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 783 . . . . . . . . . . . . 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 787 . . . . . . . . . . . . 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 19995 . . . . . . . . . . . . . . . . 17 (𝐼𝑇 ↔ (𝐼 ⊆ (Base‘𝑅) ∧ 𝐼 ≠ ∅ ∧ ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼))
6362simp3bi 1144 . . . . . . . . . . . . . . . 16 (𝐼𝑇 → ∀𝑧 ∈ (Base‘𝑅)∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6463r19.21bi 3173 . . . . . . . . . . . . . . 15 ((𝐼𝑇𝑧 ∈ (Base‘𝑅)) → ∀𝑖𝐼𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6564r19.21bi 3173 . . . . . . . . . . . . . 14 (((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) → ∀𝑗𝐼 ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6665r19.21bi 3173 . . . . . . . . . . . . 13 ((((𝐼𝑇𝑧 ∈ (Base‘𝑅)) ∧ 𝑖𝐼) ∧ 𝑗𝐼) → ((𝑧(.r𝑅)𝑖)(+g𝑅)𝑗) ∈ 𝐼)
6758, 59, 60, 61, 66syl1111anc 838 . . . . . . . . . . . 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 3917 . . . . . . . . . . 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 6981 . . . . . . . . . 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 2891 . . . . . . . . 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 725 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → 𝐹:(Base‘𝑅)⟶𝐵)
7271ffund 6496 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → Fun 𝐹)
7372ad7antr 737 . . . . . . . . . 10 ((((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) ∧ 𝑖𝐼) ∧ (𝐹𝑖) = 𝑎) → Fun 𝐹)
74 imadmrn 5909 . . . . . . . . . . . . . . . . 17 (𝐹 “ dom 𝐹) = ran 𝐹
753fdmd 6502 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝑅 RingHom 𝑆) → dom 𝐹 = (Base‘𝑅))
7675imaeq2d 5899 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ dom 𝐹) = (𝐹 “ (Base‘𝑅)))
7774, 76syl5reqr 2848 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹 “ (Base‘𝑅)) = ran 𝐹)
7877eqeq1d 2800 . . . . . . . . . . . . . . 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 2875 . . . . . . . . . . . . 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 714 . . . . . . . . . . 11 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → 𝑥 ∈ (𝐹 “ (Base‘𝑅)))
8382ad6antr 735 . . . . . . . . . 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 6713 . . . . . . . . . 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 3248 . . . . . . . 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 733 . . . . . . . . 9 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → Fun 𝐹)
88 simp-4r 783 . . . . . . . . 9 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → 𝑎 ∈ (𝐹𝐼))
89 fvelima 6713 . . . . . . . . 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 3248 . . . . . . 7 ((((((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) ∧ 𝑎 ∈ (𝐹𝐼)) ∧ 𝑏 ∈ (𝐹𝐼)) ∧ 𝑗𝐼) ∧ (𝐹𝑗) = 𝑏) → ((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9272ad3antrrr 729 . . . . . . . 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 6713 . . . . . . . 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 3248 . . . . . 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 3156 . . . 4 ((((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) ∧ 𝑥𝐵) → ∀𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼))
9998ralrimiva 3149 . . 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 19995 . . 3 ((𝐹𝐼) ∈ 𝑈 ↔ ((𝐹𝐼) ⊆ 𝐵 ∧ (𝐹𝐼) ≠ ∅ ∧ ∀𝑥𝐵𝑎 ∈ (𝐹𝐼)∀𝑏 ∈ (𝐹𝐼)((𝑥(.r𝑆)𝑎)(+g𝑆)𝑏) ∈ (𝐹𝐼)))
1026, 19, 99, 101syl3anbrc 1340 . 2 (((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵) ∧ 𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
1031023impa 1107 1 ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ ran 𝐹 = 𝐵𝐼𝑇) → (𝐹𝐼) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ⊆ wss 3882  ∅c0 4245  dom cdm 5522  ran crn 5523   “ cima 5525  Fun wfun 6323   Fn wfn 6324  ⟶wf 6325  ‘cfv 6329  (class class class)co 7142  Basecbs 16492  +gcplusg 16574  .rcmulr 16575  0gc0g 16722   GrpHom cghm 18365  Ringcrg 19308   RingHom crh 19478  LIdealclidl 19953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7451  ax-cnex 10597  ax-resscn 10598  ax-1cn 10599  ax-icn 10600  ax-addcl 10601  ax-addrcl 10602  ax-mulcl 10603  ax-mulrcl 10604  ax-mulcom 10605  ax-addass 10606  ax-mulass 10607  ax-distr 10608  ax-i2m1 10609  ax-1ne0 10610  ax-1rid 10611  ax-rnegex 10612  ax-rrecex 10613  ax-cnre 10614  ax-pre-lttri 10615  ax-pre-lttrn 10616  ax-pre-ltadd 10617  ax-pre-mulgt0 10618 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3722  df-csb 3830  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7571  df-1st 7681  df-2nd 7682  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-er 8287  df-map 8406  df-en 8508  df-dom 8509  df-sdom 8510  df-pnf 10681  df-mnf 10682  df-xr 10683  df-ltxr 10684  df-le 10685  df-sub 10876  df-neg 10877  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-ndx 16495  df-slot 16496  df-base 16498  df-sets 16499  df-ress 16500  df-plusg 16587  df-mulr 16588  df-sca 16590  df-vsca 16591  df-ip 16592  df-0g 16724  df-mgm 17861  df-sgrp 17910  df-mnd 17921  df-mhm 17965  df-grp 18115  df-minusg 18116  df-sbg 18117  df-subg 18286  df-ghm 18366  df-mgp 19251  df-ur 19263  df-ring 19310  df-rnghom 19481  df-subrg 19544  df-lmod 19647  df-lss 19715  df-sra 19955  df-rgmod 19956  df-lidl 19957 This theorem is referenced by:  rhmpreimacnlem  31300  rhmpreimacn  31301
 Copyright terms: Public domain W3C validator