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Theorem keridl 38019
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 6102 . . 3 (𝐹 “ {𝑍}) ⊆ dom 𝐹
2 eqid 2735 . . . 4 (1st𝑅) = (1st𝑅)
3 eqid 2735 . . . 4 ran (1st𝑅) = ran (1st𝑅)
4 keridl.1 . . . 4 𝐺 = (1st𝑆)
5 eqid 2735 . . . 4 ran 𝐺 = ran 𝐺
62, 3, 4, 5rngohomf 37953 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran 𝐺)
71, 6fssdm 6756 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹 “ {𝑍}) ⊆ ran (1st𝑅))
8 eqid 2735 . . . . 5 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
92, 3, 8rngo0cl 37906 . . . 4 (𝑅 ∈ RingOps → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
1093ad2ant1 1132 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (GId‘(1st𝑅)) ∈ ran (1st𝑅))
11 keridl.2 . . . . 5 𝑍 = (GId‘𝐺)
122, 8, 4, 11rngohom0 37959 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) → (𝐹‘(GId‘(1st𝑅))) = 𝑍)
13 fvex 6920 . . . . 5 (𝐹‘(GId‘(1st𝑅))) ∈ V
1413elsn 4646 . . . 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 6737 . . . 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 37956 . . . . . . . . . . . . . 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 37897 . . . . . . . . . . . . . . . 16 (𝑆 ∈ RingOps → 𝐺 ∈ GrpOp)
265, 11grpoidcl 30543 . . . . . . . . . . . . . . . 16 (𝐺 ∈ GrpOp → 𝑍 ∈ ran 𝐺)
275, 11grpolid 30545 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) → (𝑍𝐺𝑍) = 𝑍)
2825, 26, 27syl2anc2 585 . . . . . . . . . . . . . . 15 (𝑆 ∈ RingOps → (𝑍𝐺𝑍) = 𝑍)
29283ad2ant2 1133 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 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 6920 . . . . . . . . . . . . 13 (𝐹𝑥) ∈ V
3433elsn 4646 . . . . . . . . . . . 12 ((𝐹𝑥) ∈ {𝑍} ↔ (𝐹𝑥) = 𝑍)
35 fvex 6920 . . . . . . . . . . . . 13 (𝐹𝑦) ∈ V
3635elsn 4646 . . . . . . . . . . . 12 ((𝐹𝑦) ∈ {𝑍} ↔ (𝐹𝑦) = 𝑍)
3734, 36anbi12i 628 . . . . . . . . . . 11 (((𝐹𝑥) ∈ {𝑍} ∧ (𝐹𝑦) ∈ {𝑍}) ↔ ((𝐹𝑥) = 𝑍 ∧ (𝐹𝑦) = 𝑍))
38 fvex 6920 . . . . . . . . . . . 12 (𝐹‘(𝑥(1st𝑅)𝑦)) ∈ V
3938elsn 4646 . . . . . . . . . . 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 37899 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
43423expib 1121 . . . . . . . . . . 11 (𝑅 ∈ RingOps → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)))
44433ad2ant1 1132 . . . . . . . . . 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 3144 . . . 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 2735 . . . . . . . . . . . . . . . 16 (2nd𝑅) = (2nd𝑅)
602, 59, 3rngocl 37888 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
61603expb 1119 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑥 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ ran (1st𝑅))
62613ad2antl1 1184 . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . 16 (2nd𝑆) = (2nd𝑆)
662, 3, 59, 65rngohommul 37957 . . . . . . . . . . . . . . 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 37954 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹𝑧) ∈ ran 𝐺)
7311, 5, 4, 65rngorz 37910 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → ((𝐹𝑧)(2nd𝑆)𝑍) = 𝑍)
74733ad2antl2 1185 . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑧(2nd𝑅)𝑥)) = 𝑍)
78 fvex 6920 . . . . . . . . . . . . 13 (𝐹‘(𝑧(2nd𝑅)𝑥)) ∈ V
7978elsn 4646 . . . . . . . . . . . 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 37888 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
86853expb 1119 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ ran (1st𝑅))
87863ad2antl1 1184 . . . . . . . . . . . . 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 37957 . . . . . . . . . . . . . . 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 37909 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (𝐹𝑧) ∈ ran 𝐺) → (𝑍(2nd𝑆)(𝐹𝑧)) = 𝑍)
97963ad2antl2 1185 . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ (𝐹𝑥) = 𝑍)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑧)) = 𝑍)
101 fvex 6920 . . . . . . . . . . . . 13 (𝐹‘(𝑥(2nd𝑅)𝑧)) ∈ V
102101elsn 4646 . . . . . . . . . . . 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 3144 . . . . . . . 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 3144 . 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 38001 . . 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 1132 . 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 1341 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 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  {csn 4631  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GrpOpcgr 30518  GIdcgi 30519  RingOpscrngo 37881   RingOpsHom crngohom 37947  Idlcidl 37994
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  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-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-id 5583  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-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-1st 8013  df-2nd 8014  df-map 8867  df-grpo 30522  df-gid 30523  df-ginv 30524  df-ablo 30574  df-ghomOLD 37871  df-rngo 37882  df-rngohom 37950  df-idl 37997
This theorem is referenced by: (None)
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