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Theorem rngohomco 36433
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 2736 . . . . . . 7 (1st𝑆) = (1st𝑆)
2 eqid 2736 . . . . . . 7 ran (1st𝑆) = ran (1st𝑆)
3 eqid 2736 . . . . . . 7 (1st𝑇) = (1st𝑇)
4 eqid 2736 . . . . . . 7 ran (1st𝑇) = ran (1st𝑇)
51, 2, 3, 4rngohomf 36425 . . . . . 6 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
653expa 1118 . . . . 5 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
763adantl1 1166 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
87adantrl 714 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐺:ran (1st𝑆)⟶ran (1st𝑇))
9 eqid 2736 . . . . . . 7 (1st𝑅) = (1st𝑅)
10 eqid 2736 . . . . . . 7 ran (1st𝑅) = ran (1st𝑅)
119, 10, 1, 2rngohomf 36425 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
12113expa 1118 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
13123adantl3 1168 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
1413adantrr 715 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → 𝐹:ran (1st𝑅)⟶ran (1st𝑆))
15 fco 6692 . . 3 ((𝐺:ran (1st𝑆)⟶ran (1st𝑇) ∧ 𝐹:ran (1st𝑅)⟶ran (1st𝑆)) → (𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇))
168, 14, 15syl2anc 584 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹):ran (1st𝑅)⟶ran (1st𝑇))
17 eqid 2736 . . . . . . 7 (2nd𝑅) = (2nd𝑅)
18 eqid 2736 . . . . . . 7 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
1910, 17, 18rngo1cl 36398 . . . . . 6 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
20193ad2ant1 1133 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
2120adantr 481 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
22 fvco3 6940 . . . 4 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (GId‘(2nd𝑅)) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (𝐺‘(𝐹‘(GId‘(2nd𝑅)))))
2314, 21, 22syl2anc 584 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (𝐺‘(𝐹‘(GId‘(2nd𝑅)))))
24 eqid 2736 . . . . . . . . 9 (2nd𝑆) = (2nd𝑆)
25 eqid 2736 . . . . . . . . 9 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
2617, 18, 24, 25rngohom1 36427 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
27263expa 1118 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
28273adantl3 1168 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
2928adantrr 715 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
3029fveq2d 6846 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2nd𝑅)))) = (𝐺‘(GId‘(2nd𝑆))))
31 eqid 2736 . . . . . . . 8 (2nd𝑇) = (2nd𝑇)
32 eqid 2736 . . . . . . . 8 (GId‘(2nd𝑇)) = (GId‘(2nd𝑇))
3324, 25, 31, 32rngohom1 36427 . . . . . . 7 ((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
34333expa 1118 . . . . . 6 (((𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
35343adantl1 1166 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇)) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
3635adantrl 714 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑇)))
3730, 36eqtrd 2776 . . 3 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺‘(𝐹‘(GId‘(2nd𝑅)))) = (GId‘(2nd𝑇)))
3823, 37eqtrd 2776 . 2 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → ((𝐺𝐹)‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑇)))
399, 10, 1rngohomadd 36428 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦)))
4039ex 413 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
41403expa 1118 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(1st𝑅)𝑦)) = ((𝐹𝑥)(1st𝑆)(𝐹𝑦))))
42413adantl3 1168 . . . . . . . . 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 407 . . . . . . . 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 719 . . . . . . 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 6846 . . . . . 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 36426 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑥 ∈ ran (1st𝑅)) → (𝐹𝑥) ∈ ran (1st𝑆))
479, 10, 1, 2rngohomcl 36426 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹𝑦) ∈ ran (1st𝑆))
4846, 47anim12dan 619 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)))
4948ex 413 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
50493expa 1118 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
51503adantl3 1168 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆))))
5251imp 407 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → ((𝐹𝑥) ∈ ran (1st𝑆) ∧ (𝐹𝑦) ∈ ran (1st𝑆)))
5352adantlrr 719 . . . . . . 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 36428 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 1118 . . . . . . . . . 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 1166 . . . . . . . . 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 407 . . . . . . . 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 718 . . . . . . 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 591 . . . . . 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 2776 . . . . 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 36371 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
63623expb 1120 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
64633ad2antl1 1185 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅))
6564adantlr 713 . . . . . 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 6940 . . . . . . 7 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝑥(1st𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(1st𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(1st𝑅)𝑦))))
6714, 66sylan 580 . . . . . 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 591 . . . . 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 6940 . . . . . . . 8 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
7014, 69sylan 580 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑥 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
71 fvco3 6940 . . . . . . . 8 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
7214, 71sylan 580 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) ∧ 𝑦 ∈ ran (1st𝑅)) → ((𝐺𝐹)‘𝑦) = (𝐺‘(𝐹𝑦)))
7370, 72anim12dan 619 . . . . . 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 7366 . . . . . 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 2786 . . . 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 36429 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦)))
7877ex 413 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
79783expa 1118 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → ((𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝐹‘(𝑥(2nd𝑅)𝑦)) = ((𝐹𝑥)(2nd𝑆)(𝐹𝑦))))
80793adantl3 1168 . . . . . . . . 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 407 . . . . . . . 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 719 . . . . . . 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 6846 . . . . . 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 36429 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 1118 . . . . . . . . . 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 1166 . . . . . . . . 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 407 . . . . . . . 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 718 . . . . . . 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 591 . . . . . 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 2776 . . . . 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 36360 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
93923expb 1120 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
94933ad2antl1 1185 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝑥 ∈ ran (1st𝑅) ∧ 𝑦 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅))
9594adantlr 713 . . . . . 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 6940 . . . . . . 7 ((𝐹:ran (1st𝑅)⟶ran (1st𝑆) ∧ (𝑥(2nd𝑅)𝑦) ∈ ran (1st𝑅)) → ((𝐺𝐹)‘(𝑥(2nd𝑅)𝑦)) = (𝐺‘(𝐹‘(𝑥(2nd𝑅)𝑦))))
9714, 96sylan 580 . . . . . 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 591 . . . . 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 7366 . . . . . 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 2786 . . . 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 512 . . 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 3197 . 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 36424 . . . 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 1131 . . 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 481 . 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 1342 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  ran crn 5634  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  GIdcgi 29432  RingOpscrngo 36353   RngHom crnghom 36419
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-grpo 29435  df-gid 29436  df-ablo 29487  df-ass 36302  df-exid 36304  df-mgmOLD 36308  df-sgrOLD 36320  df-mndo 36326  df-rngo 36354  df-rngohom 36422
This theorem is referenced by:  rngoisoco  36441
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