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Theorem keridl 36900
Description: The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
Hypotheses
Ref Expression
keridl.1 𝐺 = (1st β€˜π‘†)
keridl.2 𝑍 = (GIdβ€˜πΊ)
Assertion
Ref Expression
keridl ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (◑𝐹 β€œ {𝑍}) ∈ (Idlβ€˜π‘…))

Proof of Theorem keridl
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvimass 6081 . . 3 (◑𝐹 β€œ {𝑍}) βŠ† dom 𝐹
2 eqid 2733 . . . 4 (1st β€˜π‘…) = (1st β€˜π‘…)
3 eqid 2733 . . . 4 ran (1st β€˜π‘…) = ran (1st β€˜π‘…)
4 keridl.1 . . . 4 𝐺 = (1st β€˜π‘†)
5 eqid 2733 . . . 4 ran 𝐺 = ran 𝐺
62, 3, 4, 5rngohomf 36834 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ 𝐹:ran (1st β€˜π‘…)⟢ran 𝐺)
71, 6fssdm 6738 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (◑𝐹 β€œ {𝑍}) βŠ† ran (1st β€˜π‘…))
8 eqid 2733 . . . . 5 (GIdβ€˜(1st β€˜π‘…)) = (GIdβ€˜(1st β€˜π‘…))
92, 3, 8rngo0cl 36787 . . . 4 (𝑅 ∈ RingOps β†’ (GIdβ€˜(1st β€˜π‘…)) ∈ ran (1st β€˜π‘…))
1093ad2ant1 1134 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (GIdβ€˜(1st β€˜π‘…)) ∈ ran (1st β€˜π‘…))
11 keridl.2 . . . . 5 𝑍 = (GIdβ€˜πΊ)
122, 8, 4, 11rngohom0 36840 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) = 𝑍)
13 fvex 6905 . . . . 5 (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) ∈ V
1413elsn 4644 . . . 4 ((πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) ∈ {𝑍} ↔ (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) = 𝑍)
1512, 14sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) ∈ {𝑍})
16 ffn 6718 . . . 4 (𝐹:ran (1st β€˜π‘…)⟢ran 𝐺 β†’ 𝐹 Fn ran (1st β€˜π‘…))
17 elpreima 7060 . . . 4 (𝐹 Fn ran (1st β€˜π‘…) β†’ ((GIdβ€˜(1st β€˜π‘…)) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((GIdβ€˜(1st β€˜π‘…)) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) ∈ {𝑍})))
186, 16, 173syl 18 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((GIdβ€˜(1st β€˜π‘…)) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((GIdβ€˜(1st β€˜π‘…)) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(GIdβ€˜(1st β€˜π‘…))) ∈ {𝑍})))
1910, 15, 18mpbir2and 712 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (GIdβ€˜(1st β€˜π‘…)) ∈ (◑𝐹 β€œ {𝑍}))
20 an4 655 . . . . . . . 8 (((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘¦) ∈ {𝑍})) ↔ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) ∧ ((πΉβ€˜π‘₯) ∈ {𝑍} ∧ (πΉβ€˜π‘¦) ∈ {𝑍})))
212, 3, 4rngohomadd 36837 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) β†’ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)𝐺(πΉβ€˜π‘¦)))
2221adantr 482 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) ∧ ((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍)) β†’ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = ((πΉβ€˜π‘₯)𝐺(πΉβ€˜π‘¦)))
23 oveq12 7418 . . . . . . . . . . . . . 14 (((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍) β†’ ((πΉβ€˜π‘₯)𝐺(πΉβ€˜π‘¦)) = (𝑍𝐺𝑍))
2423adantl 483 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) ∧ ((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍)) β†’ ((πΉβ€˜π‘₯)𝐺(πΉβ€˜π‘¦)) = (𝑍𝐺𝑍))
254rngogrpo 36778 . . . . . . . . . . . . . . . 16 (𝑆 ∈ RingOps β†’ 𝐺 ∈ GrpOp)
265, 11grpoidcl 29767 . . . . . . . . . . . . . . . 16 (𝐺 ∈ GrpOp β†’ 𝑍 ∈ ran 𝐺)
275, 11grpolid 29769 . . . . . . . . . . . . . . . 16 ((𝐺 ∈ GrpOp ∧ 𝑍 ∈ ran 𝐺) β†’ (𝑍𝐺𝑍) = 𝑍)
2825, 26, 27syl2anc2 586 . . . . . . . . . . . . . . 15 (𝑆 ∈ RingOps β†’ (𝑍𝐺𝑍) = 𝑍)
29283ad2ant2 1135 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝑍𝐺𝑍) = 𝑍)
3029ad2antrr 725 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) ∧ ((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍)) β†’ (𝑍𝐺𝑍) = 𝑍)
3122, 24, 303eqtrd 2777 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) ∧ ((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍)) β†’ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = 𝑍)
3231ex 414 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) β†’ (((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍) β†’ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = 𝑍))
33 fvex 6905 . . . . . . . . . . . . 13 (πΉβ€˜π‘₯) ∈ V
3433elsn 4644 . . . . . . . . . . . 12 ((πΉβ€˜π‘₯) ∈ {𝑍} ↔ (πΉβ€˜π‘₯) = 𝑍)
35 fvex 6905 . . . . . . . . . . . . 13 (πΉβ€˜π‘¦) ∈ V
3635elsn 4644 . . . . . . . . . . . 12 ((πΉβ€˜π‘¦) ∈ {𝑍} ↔ (πΉβ€˜π‘¦) = 𝑍)
3734, 36anbi12i 628 . . . . . . . . . . 11 (((πΉβ€˜π‘₯) ∈ {𝑍} ∧ (πΉβ€˜π‘¦) ∈ {𝑍}) ↔ ((πΉβ€˜π‘₯) = 𝑍 ∧ (πΉβ€˜π‘¦) = 𝑍))
38 fvex 6905 . . . . . . . . . . . 12 (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ V
3938elsn 4644 . . . . . . . . . . 11 ((πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍} ↔ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) = 𝑍)
4032, 37, 393imtr4g 296 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…))) β†’ (((πΉβ€˜π‘₯) ∈ {𝑍} ∧ (πΉβ€˜π‘¦) ∈ {𝑍}) β†’ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍}))
4140imdistanda 573 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) ∧ ((πΉβ€˜π‘₯) ∈ {𝑍} ∧ (πΉβ€˜π‘¦) ∈ {𝑍})) β†’ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
422, 3rngogcl 36780 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…))
43423expib 1123 . . . . . . . . . . 11 (𝑅 ∈ RingOps β†’ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…)))
44433ad2ant1 1134 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…)))
4544anim1d 612 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍}) β†’ ((π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
4641, 45syld 47 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (((π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑦 ∈ ran (1st β€˜π‘…)) ∧ ((πΉβ€˜π‘₯) ∈ {𝑍} ∧ (πΉβ€˜π‘¦) ∈ {𝑍})) β†’ ((π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
4720, 46biimtrid 241 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘¦) ∈ {𝑍})) β†’ ((π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
48 elpreima 7060 . . . . . . . . 9 (𝐹 Fn ran (1st β€˜π‘…) β†’ (π‘₯ ∈ (◑𝐹 β€œ {𝑍}) ↔ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍})))
496, 16, 483syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (π‘₯ ∈ (◑𝐹 β€œ {𝑍}) ↔ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍})))
50 elpreima 7060 . . . . . . . . 9 (𝐹 Fn ran (1st β€˜π‘…) β†’ (𝑦 ∈ (◑𝐹 β€œ {𝑍}) ↔ (𝑦 ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘¦) ∈ {𝑍})))
516, 16, 503syl 18 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (𝑦 ∈ (◑𝐹 β€œ {𝑍}) ↔ (𝑦 ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘¦) ∈ {𝑍})))
5249, 51anbi12d 632 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯ ∈ (◑𝐹 β€œ {𝑍}) ∧ 𝑦 ∈ (◑𝐹 β€œ {𝑍})) ↔ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍}) ∧ (𝑦 ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘¦) ∈ {𝑍}))))
53 elpreima 7060 . . . . . . . 8 (𝐹 Fn ran (1st β€˜π‘…) β†’ ((π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
546, 16, 533syl 18 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((π‘₯(1st β€˜π‘…)𝑦) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(1st β€˜π‘…)𝑦)) ∈ {𝑍})))
5547, 52, 543imtr4d 294 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯ ∈ (◑𝐹 β€œ {𝑍}) ∧ 𝑦 ∈ (◑𝐹 β€œ {𝑍})) β†’ (π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍})))
5655impl 457 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ (◑𝐹 β€œ {𝑍})) ∧ 𝑦 ∈ (◑𝐹 β€œ {𝑍})) β†’ (π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}))
5756ralrimiva 3147 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ (◑𝐹 β€œ {𝑍})) β†’ βˆ€π‘¦ ∈ (◑𝐹 β€œ {𝑍})(π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}))
5834anbi2i 624 . . . . . . 7 ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍}) ↔ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍))
59 eqid 2733 . . . . . . . . . . . . . . . 16 (2nd β€˜π‘…) = (2nd β€˜π‘…)
602, 59, 3rngocl 36769 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st β€˜π‘…) ∧ π‘₯ ∈ ran (1st β€˜π‘…)) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…))
61603expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝑧 ∈ ran (1st β€˜π‘…) ∧ π‘₯ ∈ ran (1st β€˜π‘…))) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…))
62613ad2antl1 1186 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1st β€˜π‘…) ∧ π‘₯ ∈ ran (1st β€˜π‘…))) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…))
6362anass1rs 654 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ ran (1st β€˜π‘…)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…))
6463adantlrr 720 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…))
65 eqid 2733 . . . . . . . . . . . . . . . 16 (2nd β€˜π‘†) = (2nd β€˜π‘†)
662, 3, 59, 65rngohommul 36838 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑧 ∈ ran (1st β€˜π‘…) ∧ π‘₯ ∈ ran (1st β€˜π‘…))) β†’ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)))
6766anass1rs 654 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ ran (1st β€˜π‘…)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)))
6867adantlrr 720 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)))
69 oveq2 7417 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘₯) = 𝑍 β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍))
7069adantl 483 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍))
7170ad2antlr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)(πΉβ€˜π‘₯)) = ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍))
722, 3, 4, 5rngohomcl 36835 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜π‘§) ∈ ran 𝐺)
7311, 5, 4, 65rngorz 36791 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (πΉβ€˜π‘§) ∈ ran 𝐺) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍) = 𝑍)
74733ad2antl2 1187 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (πΉβ€˜π‘§) ∈ ran 𝐺) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍) = 𝑍)
7572, 74syldan 592 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍) = 𝑍)
7675adantlr 714 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((πΉβ€˜π‘§)(2nd β€˜π‘†)𝑍) = 𝑍)
7768, 71, 763eqtrd 2777 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) = 𝑍)
78 fvex 6905 . . . . . . . . . . . . 13 (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ V
7978elsn 4644 . . . . . . . . . . . 12 ((πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ {𝑍} ↔ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) = 𝑍)
8077, 79sylibr 233 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ {𝑍})
81 elpreima 7060 . . . . . . . . . . . . 13 (𝐹 Fn ran (1st β€˜π‘…) β†’ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ {𝑍})))
826, 16, 813syl 18 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ {𝑍})))
8382ad2antrr 725 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(𝑧(2nd β€˜π‘…)π‘₯)) ∈ {𝑍})))
8464, 80, 83mpbir2and 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}))
852, 59, 3rngocl 36769 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…))
86853expb 1121 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑧 ∈ ran (1st β€˜π‘…))) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…))
87863ad2antl1 1186 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑧 ∈ ran (1st β€˜π‘…))) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…))
8887anassrs 469 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ ran (1st β€˜π‘…)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…))
8988adantlrr 720 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…))
902, 3, 59, 65rngohommul 36838 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ 𝑧 ∈ ran (1st β€˜π‘…))) β†’ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)))
9190anassrs 469 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ ran (1st β€˜π‘…)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)))
9291adantlrr 720 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) = ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)))
93 oveq1 7416 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘₯) = 𝑍 β†’ ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)) = (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)))
9493adantl 483 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍) β†’ ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)) = (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)))
9594ad2antlr 726 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((πΉβ€˜π‘₯)(2nd β€˜π‘†)(πΉβ€˜π‘§)) = (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)))
9611, 5, 4, 65rngolz 36790 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ RingOps ∧ (πΉβ€˜π‘§) ∈ ran 𝐺) β†’ (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)) = 𝑍)
97963ad2antl2 1187 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (πΉβ€˜π‘§) ∈ ran 𝐺) β†’ (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)) = 𝑍)
9872, 97syldan 592 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)) = 𝑍)
9998adantlr 714 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (𝑍(2nd β€˜π‘†)(πΉβ€˜π‘§)) = 𝑍)
10092, 95, 993eqtrd 2777 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) = 𝑍)
101 fvex 6905 . . . . . . . . . . . . 13 (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ V
102101elsn 4644 . . . . . . . . . . . 12 ((πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ {𝑍} ↔ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) = 𝑍)
103100, 102sylibr 233 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ {𝑍})
104 elpreima 7060 . . . . . . . . . . . . 13 (𝐹 Fn ran (1st β€˜π‘…) β†’ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ {𝑍})))
1056, 16, 1043syl 18 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ {𝑍})))
106105ad2antrr 725 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}) ↔ ((π‘₯(2nd β€˜π‘…)𝑧) ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜(π‘₯(2nd β€˜π‘…)𝑧)) ∈ {𝑍})))
10789, 103, 106mpbir2and 712 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))
10884, 107jca 513 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) ∧ 𝑧 ∈ ran (1st β€˜π‘…)) β†’ ((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍})))
109108ralrimiva 3147 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍)) β†’ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍})))
110109ex 414 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) = 𝑍) β†’ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))
11158, 110biimtrid 241 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((π‘₯ ∈ ran (1st β€˜π‘…) ∧ (πΉβ€˜π‘₯) ∈ {𝑍}) β†’ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))
11249, 111sylbid 239 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (π‘₯ ∈ (◑𝐹 β€œ {𝑍}) β†’ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))
113112imp 408 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ (◑𝐹 β€œ {𝑍})) β†’ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍})))
11457, 113jca 513 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ π‘₯ ∈ (◑𝐹 β€œ {𝑍})) β†’ (βˆ€π‘¦ ∈ (◑𝐹 β€œ {𝑍})(π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))
115114ralrimiva 3147 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ βˆ€π‘₯ ∈ (◑𝐹 β€œ {𝑍})(βˆ€π‘¦ ∈ (◑𝐹 β€œ {𝑍})(π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))
1162, 59, 3, 8isidl 36882 . . 3 (𝑅 ∈ RingOps β†’ ((◑𝐹 β€œ {𝑍}) ∈ (Idlβ€˜π‘…) ↔ ((◑𝐹 β€œ {𝑍}) βŠ† ran (1st β€˜π‘…) ∧ (GIdβ€˜(1st β€˜π‘…)) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘₯ ∈ (◑𝐹 β€œ {𝑍})(βˆ€π‘¦ ∈ (◑𝐹 β€œ {𝑍})(π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))))
1171163ad2ant1 1134 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ ((◑𝐹 β€œ {𝑍}) ∈ (Idlβ€˜π‘…) ↔ ((◑𝐹 β€œ {𝑍}) βŠ† ran (1st β€˜π‘…) ∧ (GIdβ€˜(1st β€˜π‘…)) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘₯ ∈ (◑𝐹 β€œ {𝑍})(βˆ€π‘¦ ∈ (◑𝐹 β€œ {𝑍})(π‘₯(1st β€˜π‘…)𝑦) ∈ (◑𝐹 β€œ {𝑍}) ∧ βˆ€π‘§ ∈ ran (1st β€˜π‘…)((𝑧(2nd β€˜π‘…)π‘₯) ∈ (◑𝐹 β€œ {𝑍}) ∧ (π‘₯(2nd β€˜π‘…)𝑧) ∈ (◑𝐹 β€œ {𝑍}))))))
1187, 19, 115, 117mpbir3and 1343 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) β†’ (◑𝐹 β€œ {𝑍}) ∈ (Idlβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949  {csn 4629  β—‘ccnv 5676  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GrpOpcgr 29742  GIdcgi 29743  RingOpscrngo 36762   RngHom crnghom 36828  Idlcidl 36875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822  df-grpo 29746  df-gid 29747  df-ginv 29748  df-ablo 29798  df-ghomOLD 36752  df-rngo 36763  df-rngohom 36831  df-idl 36878
This theorem is referenced by: (None)
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