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Theorem rngohomco 32943
Description: The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngohomco (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))

Proof of Theorem rngohomco
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . . 7 (1st𝑆) = (1st𝑆)
2 eqid 2610 . . . . . . 7 ran (1st𝑆) = ran (1st𝑆)
3 eqid 2610 . . . . . . 7 (1st𝑇) = (1st𝑇)
4 eqid 2610 . . . . . . 7 ran (1st𝑇) = ran (1st𝑇)
51, 2, 3, 4rngohomf 32935 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
653expa 1257 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
763adantl1 1210 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
87adantrl 748 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
9 eqid 2610 . . . . . . 7 (1st𝑅) = (1st𝑅)
10 eqid 2610 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
119, 10, 1, 2rngohomf 32935 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
12113expa 1257 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
13123adantl3 1212 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
1413adantrr 749 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
15 fco 5971 . . 3 ((𝐺:ran (1st𝑆)⟶ran (1st𝑇) ∧ 𝐹:ran (1st𝑅)⟶ran (1st𝑆)) → (𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇))
168, 14, 15syl2anc 691 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇))
17 eqid 2610 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
18 eqid 2610 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
1910, 17, 18rngo1cl 32908 . . . . . 6 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
20193ad2ant1 1075 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
2120adantr 480 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
22 fvco3 6185 . . . 4 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (GId‘(2nd𝑅)) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (𝐺‘(𝐹‘(GId‘(2nd𝑅)))))
2314, 21, 22syl2anc 691 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (𝐺‘(𝐹‘(GId‘(2nd𝑅)))))
24 eqid 2610 . . . . . . . . 9 (2nd𝑆) = (2nd𝑆)
25 eqid 2610 . . . . . . . . 9 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
2617, 18, 24, 25rngohom1 32937 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
27263expa 1257 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
28273adantl3 1212 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
2928adantrr 749 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
3029fveq2d 6107 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2nd𝑅)))) = (𝐺‘(GId‘(2nd𝑆))))
31 eqid 2610 . . . . . . . 8 (2nd𝑇) = (2nd𝑇)
32 eqid 2610 . . . . . . . 8 (GId‘(2nd𝑇)) = (GId‘(2nd𝑇))
3324, 25, 31, 32rngohom1 32937 . . . . . . 7 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
34333expa 1257 . . . . . 6 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
35343adantl1 1210 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
3635adantrl 748 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
3730, 36eqtrd 2644 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2nd𝑅)))) = (GId‘(2nd𝑇)))
3823, 37eqtrd 2644 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑇)))
399, 10, 1rngohomadd 32938 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)))
4039ex 449 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
41403expa 1257 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
42413adantl3 1212 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
4342imp 444 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)))
4443adantlrr 753 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)))
4544fveq2d 6107 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))) = (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
469, 10, 1, 2rngohomcl 32936 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) → (𝐹𝑥) ∈ ran (1st𝑆))
479, 10, 1, 2rngohomcl 32936 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹𝑦) ∈ ran (1st𝑆))
4846, 47anim12da 32676 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)))
4948ex 449 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
50493expa 1257 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
51503adantl3 1212 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
5251imp 444 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)))
5352adantlrr 753 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)))
541, 2, 3rngohomadd 32938 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
5554ex 449 . . . . . . . . . . 11 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦)))))
56553expa 1257 . . . . . . . . . 10 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦)))))
57563adantl1 1210 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦)))))
5857imp 444 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
5958adantlrl 752 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
6053, 59syldan 486 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘((𝐹𝑥)(1st𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
6145, 60eqtrd 2644 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
629, 10rngogcl 32881 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
63623expb 1258 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
64633ad2antl1 1216 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
6564adantlr 747 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
66 fvco3 6185 . . . . . . 7 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))))
6714, 66sylan 487 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))))
6865, 67syldan 486 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))))
69 fvco3 6185 . . . . . . . 8 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
7014, 69sylan 487 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑥 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
71 fvco3 6185 . . . . . . . 8 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
7214, 71sylan 487 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
7370, 72anim12da 32676 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)) ∧ ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦))))
74 oveq12 6558 . . . . . 6 ((((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)) ∧ ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦))) → (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
7573, 74syl 17 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) = ((𝐺‘(𝐹𝑥))(1st𝑇)(𝐺‘(𝐹𝑦))))
7661, 68, 753eqtr4d 2654 . . . 4 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)))
779, 10, 17, 24rngohommul 32939 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))
7877ex 449 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
79783expa 1257 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
80793adantl3 1212 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
8180imp 444 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))
8281adantlrr 753 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))
8382fveq2d 6107 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))) = (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
841, 2, 24, 31rngohommul 32939 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
8584ex 449 . . . . . . . . . . 11 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦)))))
86853expa 1257 . . . . . . . . . 10 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦)))))
87863adantl1 1210 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦)))))
8887imp 444 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
8988adantlrl 752 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
9053, 89syldan 486 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘((𝐹𝑥)(2nd𝑆)(𝐹𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
9183, 90eqtrd 2644 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
929, 17, 10rngocl 32870 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
93923expb 1258 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
94933ad2antl1 1216 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
9594adantlr 747 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
96 fvco3 6185 . . . . . . 7 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))))
9714, 96sylan 487 . . . . . 6 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))))
9895, 97syldan 486 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))))
99 oveq12 6558 . . . . . 6 ((((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)) ∧ ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦))) → (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦)) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
10073, 99syl 17 . . . . 5 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦)) = ((𝐺‘(𝐹𝑥))(2nd𝑇)(𝐺‘(𝐹𝑦))))
10191, 98, 1003eqtr4d 2654 . . . 4 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦)))
10276, 101jca 553 . . 3 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) ∧ ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦))))
103102ralrimivva 2954 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) ∧ ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦))))
1049, 17, 10, 18, 3, 31, 4, 32isrngohom 32934 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇) ∧ ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑇)) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) ∧ ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦))))))
1051043adant2 1073 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇) ∧ ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑇)) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) ∧ ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦))))))
106105adantr 480 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺𝐹) ∈ (𝑅 RngHom 𝑇) ↔ ((𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇) ∧ ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑇)) ∧ ∀𝑥 ∈ ran (1st𝑅)∀𝑦 ∈ ran (1st𝑅)(((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(1st𝑇)((𝐺𝐹)‘𝑦)) ∧ ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (((𝐺𝐹)‘𝑥)(2nd𝑇)((𝐺𝐹)‘𝑦))))))
10716, 38, 103, 106mpbir3and 1238 1 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  ran crn 5039  ccom 5042  wf 5800  cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863   RngHom crnghom 32929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-grpo 26731  df-gid 26732  df-ablo 26783  df-ass 32812  df-exid 32814  df-mgmOLD 32818  df-sgrOLD 32830  df-mndo 32836  df-rngo 32864  df-rngohom 32932
This theorem is referenced by:  rngoisoco  32951
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