Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  keridl Structured version   Visualization version   GIF version

Theorem keridl 35370
Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
keridl.1 𝐺 = (1st𝑆)
keridl.2 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
keridl ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 “ {𝑍}) ∈ (Idl‘𝑅))

Proof of Theorem keridl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 5930 . . 3 (𝐹 “ {𝑍}) ⊆ dom 𝐹
2 eqid 2824 . . . 4 (1st𝑅) = (1st𝑅)
3 eqid 2824 . . . 4 ran (1st𝑅) = ran (1st𝑅)
4 keridl.1 . . . 4 𝐺 = (1st𝑆)
5 eqid 2824 . . . 4 ran 𝐺 = ran 𝐺
62, 3, 4, 5rngohomf 35304 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran 𝐺)
71, 6fssdm 6511 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 “ {𝑍}) ⊆ ran (1st𝑅))
8 eqid 2824 . . . . 5 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
92, 3, 8rngo0cl 35257 . . . 4 (𝑅 ∈ RingOps → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
1093ad2ant1 1130 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
11 keridl.2 . . . . 5 𝑍 = (GId‘𝐺)
122, 8, 4, 11rngohom0 35310 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1st𝑅))) = 𝑍)
13 fvex 6664 . . . . 5 (𝐹‘(GId‘(1st𝑅))) ∈ V
1413elsn 4563 . . . 4 ((𝐹‘(GId‘(1st𝑅))) ∈ {𝑍} ↔ (𝐹‘(GId‘(1st𝑅))) = 𝑍)
1512, 14sylibr 237 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(1st𝑅))) ∈ {𝑍})
16 ffn 6495 . . . 4 (𝐹:ran (1st𝑅)⟶ran 𝐺𝐹 Fn ran (1st𝑅))
17 elpreima 6809 . . . 4 (𝐹 Fn ran (1st𝑅) → ((GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}) ↔ ((GId‘(1st𝑅)) ∈ ran (1st𝑅) ∧ (𝐹‘(GId‘(1st𝑅))) ∈ {𝑍})))
186, 16, 173syl 18 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}) ↔ ((GId‘(1st𝑅)) ∈ ran (1st𝑅) ∧ (𝐹‘(GId‘(1st𝑅))) ∈ {𝑍})))
1910, 15, 18mpbir2and 712 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}))
20 an4 655 . . . . . . . 8 (((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})) ↔ ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})))
212, 3, 4rngohomadd 35307 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)𝐺(𝐹𝑦)))
2221adantr 484 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)𝐺(𝐹𝑦)))
23 oveq12 7147 . . . . . . . . . . . . . 14 (((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑥)𝐺(𝐹𝑦)) = (𝑍𝐺𝑍))
2423adantl 485 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = (𝑍𝐺𝑍))
254rngogrpo 35248 . . . . . . . . . . . . . . . 16 (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
265, 11grpoidcl 28286 . . . . . . . . . . . . . . . 16 (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
275, 11grpolid 28288 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
2825, 26, 27syl2anc2 588 . . . . . . . . . . . . . . 15 (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
29283ad2ant2 1131 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
3029ad2antrr 725 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝑍𝐺𝑍) = 𝑍)
3122, 24, 303eqtrd 2863 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍)
3231ex 416 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍) → (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍))
33 fvex 6664 . . . . . . . . . . . . 13 (𝐹𝑥) ∈ V
3433elsn 4563 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ {𝑍} ↔ (𝐹𝑥) = 𝑍)
35 fvex 6664 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
3635elsn 4563 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ {𝑍} ↔ (𝐹𝑦) = 𝑍)
3734, 36anbi12i 629 . . . . . . . . . . 11 (((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍}) ↔ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍))
38 fvex 6664 . . . . . . . . . . . 12 (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ V
3938elsn 4563 . . . . . . . . . . 11 ((𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍} ↔ (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍)
4032, 37, 393imtr4g 299 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍}) → (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍}))
4140imdistanda 575 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
422, 3rngogcl 35250 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
43423expib 1119 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)))
44433ad2ant1 1130 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)))
4544anim1d 613 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍}) → ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
4641, 45syld 47 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
4720, 46syl5bi 245 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
48 elpreima 6809 . . . . . . . . 9 (𝐹 Fn ran (1st𝑅) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍})))
496, 16, 483syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍})))
50 elpreima 6809 . . . . . . . . 9 (𝐹 Fn ran (1st𝑅) → (𝑦 ∈ (𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})))
516, 16, 503syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑦 ∈ (𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})))
5249, 51anbi12d 633 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (𝐹 “ {𝑍}) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) ↔ ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍}))))
53 elpreima 6809 . . . . . . . 8 (𝐹 Fn ran (1st𝑅) → ((𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
546, 16, 533syl 18 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
5547, 52, 543imtr4d 297 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ (𝐹 “ {𝑍}) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) → (𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍})))
5655impl 459 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) → (𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}))
5756ralrimiva 3176 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) → ∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}))
5834anbi2i 625 . . . . . . 7 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍))
59 eqid 2824 . . . . . . . . . . . . . . . 16 (2nd𝑅) = (2nd𝑅)
602, 59, 3rngocl 35239 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
61603expb 1117 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
62613ad2antl1 1182 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
6362anass1rs 654 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
6463adantlrr 720 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
65 eqid 2824 . . . . . . . . . . . . . . . 16 (2nd𝑆) = (2nd𝑆)
662, 3, 59, 65rngohommul 35308 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
6766anass1rs 654 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
6867adantlrr 720 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
69 oveq2 7146 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = 𝑍 → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
7069adantl 485 . . . . . . . . . . . . . 14 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
7170ad2antlr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
722, 3, 4, 5rngohomcl 35305 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹𝑧) ∈ ran 𝐺)
7311, 5, 4, 65rngorz 35261 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
74733ad2antl2 1183 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹𝑧) ∈ ran 𝐺) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7572, 74syldan 594 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7675adantlr 714 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7768, 71, 763eqtrd 2863 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = 𝑍)
78 fvex 6664 . . . . . . . . . . . . 13 (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ V
7978elsn 4563 . . . . . . . . . . . 12 ((𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍} ↔ (𝐹‘(𝑧(2nd𝑅)𝑥)) = 𝑍)
8077, 79sylibr 237 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})
81 elpreima 6809 . . . . . . . . . . . . 13 (𝐹 Fn ran (1st𝑅) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})))
826, 16, 813syl 18 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})))
8382ad2antrr 725 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})))
8464, 80, 83mpbir2and 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}))
852, 59, 3rngocl 35239 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
86853expb 1117 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
87863ad2antl1 1182 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
8887anassrs 471 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
8988adantlrr 720 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
902, 3, 59, 65rngohommul 35308 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
9190anassrs 471 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
9291adantlrr 720 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
93 oveq1 7145 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = 𝑍 → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9493adantl 485 . . . . . . . . . . . . . 14 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9594ad2antlr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9611, 5, 4, 65rngolz 35260 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
97963ad2antl2 1183 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹𝑧) ∈ ran 𝐺) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
9872, 97syldan 594 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
9998adantlr 714 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
10092, 95, 993eqtrd 2863 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = 𝑍)
101 fvex 6664 . . . . . . . . . . . . 13 (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ V
102101elsn 4563 . . . . . . . . . . . 12 ((𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍} ↔ (𝐹‘(𝑥(2nd𝑅)𝑧)) = 𝑍)
103100, 102sylibr 237 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})
104 elpreima 6809 . . . . . . . . . . . . 13 (𝐹 Fn ran (1st𝑅) → ((𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})))
1056, 16, 1043syl 18 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})))
106105ad2antrr 725 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})))
10789, 103, 106mpbir2and 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))
10884, 107jca 515 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
109108ralrimiva 3176 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
110109ex 416 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
11158, 110syl5bi 245 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
11249, 111sylbid 243 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝑥 ∈ (𝐹 “ {𝑍}) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
113112imp 410 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
11457, 113jca 515 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) → (∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
115114ralrimiva 3176 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ∀𝑥 ∈ (𝐹 “ {𝑍})(∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
1162, 59, 3, 8isidl 35352 . . 3 (𝑅 ∈ RingOps → ((𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((𝐹 “ {𝑍}) ⊆ ran (1st𝑅) ∧ (GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (𝐹 “ {𝑍})(∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))))
1171163ad2ant1 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝐹 “ {𝑍}) ∈ (Idl‘𝑅) ↔ ((𝐹 “ {𝑍}) ⊆ ran (1st𝑅) ∧ (GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑥 ∈ (𝐹 “ {𝑍})(∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))))
1187, 19, 115, 117mpbir3and 1339 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹 “ {𝑍}) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  wss 3918  {csn 4548  ccnv 5535  ran crn 5537  cima 5539   Fn wfn 6331  wf 6332  cfv 6336  (class class class)co 7138  1st c1st 7670  2nd c2nd 7671  GrpOpcgr 28261  GIdcgi 28262  RingOpscrngo 35232   RngHom crnghom 35298  Idlcidl 35345
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-1st 7672  df-2nd 7673  df-map 8391  df-grpo 28265  df-gid 28266  df-ginv 28267  df-ablo 28317  df-ghomOLD 35222  df-rngo 35233  df-rngohom 35301  df-idl 35348
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator