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Theorem keridl 38039
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 RingOpsHom 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 6100 . . 3 (𝐹 “ {𝑍}) ⊆ dom 𝐹
2 eqid 2737 . . . 4 (1st𝑅) = (1st𝑅)
3 eqid 2737 . . . 4 ran (1st𝑅) = ran (1st𝑅)
4 keridl.1 . . . 4 𝐺 = (1st𝑆)
5 eqid 2737 . . . 4 ran 𝐺 = ran 𝐺
62, 3, 4, 5rngohomf 37973 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran 𝐺)
71, 6fssdm 6755 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹 “ {𝑍}) ⊆ ran (1st𝑅))
8 eqid 2737 . . . . 5 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
92, 3, 8rngo0cl 37926 . . . 4 (𝑅 ∈ RingOps → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
1093ad2ant1 1134 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
11 keridl.2 . . . . 5 𝑍 = (GId‘𝐺)
122, 8, 4, 11rngohom0 37979 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1st𝑅))) = 𝑍)
13 fvex 6919 . . . . 5 (𝐹‘(GId‘(1st𝑅))) ∈ V
1413elsn 4641 . . . 4 ((𝐹‘(GId‘(1st𝑅))) ∈ {𝑍} ↔ (𝐹‘(GId‘(1st𝑅))) = 𝑍)
1512, 14sylibr 234 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1st𝑅))) ∈ {𝑍})
16 ffn 6736 . . . 4 (𝐹:ran (1st𝑅)⟶ran 𝐺𝐹 Fn ran (1st𝑅))
17 elpreima 7078 . . . 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 RingOpsHom 𝑆)) → ((GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}) ↔ ((GId‘(1st𝑅)) ∈ ran (1st𝑅) ∧ (𝐹‘(GId‘(1st𝑅))) ∈ {𝑍})))
1910, 15, 18mpbir2and 713 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1st𝑅)) ∈ (𝐹 “ {𝑍}))
20 an4 656 . . . . . . . 8 (((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})) ↔ ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})))
212, 3, 4rngohomadd 37976 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)𝐺(𝐹𝑦)))
2221adantr 480 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)𝐺(𝐹𝑦)))
23 oveq12 7440 . . . . . . . . . . . . . 14 (((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍) → ((𝐹𝑥)𝐺(𝐹𝑦)) = (𝑍𝐺𝑍))
2423adantl 481 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → ((𝐹𝑥)𝐺(𝐹𝑦)) = (𝑍𝐺𝑍))
254rngogrpo 37917 . . . . . . . . . . . . . . . 16 (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
265, 11grpoidcl 30533 . . . . . . . . . . . . . . . 16 (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
275, 11grpolid 30535 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
2825, 26, 27syl2anc2 585 . . . . . . . . . . . . . . 15 (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
29283ad2ant2 1135 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝑍𝐺𝑍) = 𝑍)
3029ad2antrr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝑍𝐺𝑍) = 𝑍)
3122, 24, 303eqtrd 2781 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) ∧ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍)
3231ex 412 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍) → (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍))
33 fvex 6919 . . . . . . . . . . . . 13 (𝐹𝑥) ∈ V
3433elsn 4641 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ {𝑍} ↔ (𝐹𝑥) = 𝑍)
35 fvex 6919 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
3635elsn 4641 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ {𝑍} ↔ (𝐹𝑦) = 𝑍)
3734, 36anbi12i 628 . . . . . . . . . . 11 (((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍}) ↔ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍))
38 fvex 6919 . . . . . . . . . . . 12 (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ V
3938elsn 4641 . . . . . . . . . . 11 ((𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍} ↔ (𝐹‘(𝑥(1st𝑅)𝑦)) = 𝑍)
4032, 37, 393imtr4g 296 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍}) → (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍}))
4140imdistanda 571 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
422, 3rngogcl 37919 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
43423expib 1123 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)))
44433ad2ant1 1134 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)))
4544anim1d 611 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 RingOpsHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) ∧ ((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
4720, 46biimtrid 242 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})) → ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
48 elpreima 7078 . . . . . . . . 9 (𝐹 Fn ran (1st𝑅) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍})))
496, 16, 483syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝑥 ∈ (𝐹 “ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍})))
50 elpreima 7078 . . . . . . . . 9 (𝐹 Fn ran (1st𝑅) → (𝑦 ∈ (𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})))
516, 16, 503syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝑦 ∈ (𝐹 “ {𝑍}) ↔ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍})))
5249, 51anbi12d 632 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ (𝐹 “ {𝑍}) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) ↔ ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ {𝑍}))))
53 elpreima 7078 . . . . . . . 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 RingOpsHom 𝑆)) → ((𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ {𝑍})))
5547, 52, 543imtr4d 294 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ (𝐹 “ {𝑍}) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) → (𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍})))
5655impl 455 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) ∧ 𝑦 ∈ (𝐹 “ {𝑍})) → (𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}))
5756ralrimiva 3146 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) → ∀𝑦 ∈ (𝐹 “ {𝑍})(𝑥(1st𝑅)𝑦) ∈ (𝐹 “ {𝑍}))
5834anbi2i 623 . . . . . . 7 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) ↔ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍))
59 eqid 2737 . . . . . . . . . . . . . . . 16 (2nd𝑅) = (2nd𝑅)
602, 59, 3rngocl 37908 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
61603expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
62613ad2antl1 1186 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
6362anass1rs 655 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
6463adantlrr 721 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
65 eqid 2737 . . . . . . . . . . . . . . . 16 (2nd𝑆) = (2nd𝑆)
662, 3, 59, 65rngohommul 37977 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
6766anass1rs 655 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
6867adantlrr 721 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)))
69 oveq2 7439 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = 𝑍 → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
7069adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
7170ad2antlr 727 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)(𝐹𝑥)) = ((𝐹𝑧)(2nd𝑆)𝑍))
722, 3, 4, 5rngohomcl 37974 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹𝑧) ∈ ran 𝐺)
7311, 5, 4, 65rngorz 37930 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
74733ad2antl2 1187 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹𝑧) ∈ ran 𝐺) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7572, 74syldan 591 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7675adantlr 715 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
7768, 71, 763eqtrd 2781 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = 𝑍)
78 fvex 6919 . . . . . . . . . . . . 13 (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ V
7978elsn 4641 . . . . . . . . . . . 12 ((𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍} ↔ (𝐹‘(𝑧(2nd𝑅)𝑥)) = 𝑍)
8077, 79sylibr 234 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})
81 elpreima 7078 . . . . . . . . . . . . 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 RingOpsHom 𝑆)) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ↔ ((𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ {𝑍})))
8382ad2antrr 726 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}))
852, 59, 3rngocl 37908 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
86853expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
87863ad2antl1 1186 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
8887anassrs 467 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
8988adantlrr 721 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
902, 3, 59, 65rngohommul 37977 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
9190anassrs 467 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
9291adantlrr 721 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)))
93 oveq1 7438 . . . . . . . . . . . . . . 15 ((𝐹𝑥) = 𝑍 → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9493adantl 481 . . . . . . . . . . . . . 14 ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9594ad2antlr 727 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝐹𝑥)(2nd𝑆)(𝐹𝑧)) = (𝑍(2nd𝑆)(𝐹𝑧)))
9611, 5, 4, 65rngolz 37929 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
97963ad2antl2 1187 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝐹𝑧) ∈ ran 𝐺) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
9872, 97syldan 591 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
9998adantlr 715 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
10092, 95, 993eqtrd 2781 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = 𝑍)
101 fvex 6919 . . . . . . . . . . . . 13 (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ V
102101elsn 4641 . . . . . . . . . . . 12 ((𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍} ↔ (𝐹‘(𝑥(2nd𝑅)𝑧)) = 𝑍)
103100, 102sylibr 234 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})
104 elpreima 7078 . . . . . . . . . . . . 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 RingOpsHom 𝑆)) → ((𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}) ↔ ((𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅) ∧ (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ {𝑍})))
106105ad2antrr 726 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))
10884, 107jca 511 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
109108ralrimiva 3146 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
110109ex 412 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
11158, 110biimtrid 242 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) ∈ {𝑍}) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
11249, 111sylbid 240 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝑥 ∈ (𝐹 “ {𝑍}) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍}))))
113112imp 406 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑥 ∈ (𝐹 “ {𝑍})) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ (𝐹 “ {𝑍}) ∧ (𝑥(2nd𝑅)𝑧) ∈ (𝐹 “ {𝑍})))
11457, 113jca 511 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 3146 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 38021 . . 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 1134 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 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 1343 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹 “ {𝑍}) ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  {csn 4626  ccnv 5684  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  GrpOpcgr 30508  GIdcgi 30509  RingOpscrngo 37901   RingOpsHom crngohom 37967  Idlcidl 38014
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3062  df-rex 3071  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-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-id 5578  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-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-1st 8014  df-2nd 8015  df-map 8868  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-ghomOLD 37891  df-rngo 37902  df-rngohom 37970  df-idl 38017
This theorem is referenced by: (None)
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