Theorem mrsubco 35703

Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypothesis

Ref	Expression
mrsubco.s	⊢ 𝑆 = (mRSubst‘𝑇)

Assertion

Ref	Expression
mrsubco	⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Proof of Theorem mrsubco

Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubco.s	. . . . 5 ⊢ 𝑆 = (mRSubst‘𝑇)
2		eqid 2736	. . . . 5 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
3	1, 2	mrsubf 35699	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
4	3	adantr 480	. . 3 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
5	1, 2	mrsubf 35699	. . . 4 ⊢ (𝐺 ∈ ran 𝑆 → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
6	5	adantl 481	. . 3 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7		fco 6692	. . 3 ⊢ ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
8	4, 6, 7	syl2anc 585	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
9	6	adantr 480	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
10		eldifi 4071	. . . . . . . . 9 ⊢ (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11		elun1 4122	. . . . . . . . 9 ⊢ (𝑐 ∈ (mCN‘𝑇) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
13	12	adantl 481	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
14	13	s1cld 14566	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15		n0i 4280	. . . . . . . . . 10 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
16	1	rnfvprc 6834	. . . . . . . . . 10 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
17	15, 16	nsyl2 141	. . . . . . . . 9 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
18	17	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
19	18	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑇 ∈ V)
20		eqid 2736	. . . . . . . 8 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
21		eqid 2736	. . . . . . . 8 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
22	20, 21, 2	mrexval 35683	. . . . . . 7 ⊢ (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
23	19, 22	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24	14, 23	eleqtrrd 2839	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25		fvco3 6939	. . . . 5 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
26	9, 24, 25	syl2anc 585	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
27	1, 2, 21, 20	mrsubcn 35701	. . . . . 6 ⊢ ((𝐺 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
28	27	adantll 715	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
29	28	fveq2d 6844	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
30	1, 2, 21, 20	mrsubcn 35701	. . . . 5 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
31	30	adantlr 716	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
32	26, 29, 31	3eqtrd 2775	. . 3 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
33	32	ralrimiva 3129	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
34	1, 2	mrsubccat 35700	. . . . . . . 8 ⊢ ((𝐺 ∈ ran 𝑆 ∧ 𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
35	34	3expb 1121	. . . . . . 7 ⊢ ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
36	35	adantll 715	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
37	36	fveq2d 6844	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺‘𝑥) ++ (𝐺‘𝑦))))
38		simpll 767	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐹 ∈ ran 𝑆)
39	6	adantr 480	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
40		simprl 771	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
41	39, 40	ffvelcdmd 7037	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘𝑥) ∈ (mREx‘𝑇))
42		simprr 773	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
43	39, 42	ffvelcdmd 7037	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘𝑦) ∈ (mREx‘𝑇))
44	1, 2	mrsubccat 35700	. . . . . 6 ⊢ ((𝐹 ∈ ran 𝑆 ∧ (𝐺‘𝑥) ∈ (mREx‘𝑇) ∧ (𝐺‘𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺‘𝑥) ++ (𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥)) ++ (𝐹‘(𝐺‘𝑦))))
45	38, 41, 43, 44	syl3anc 1374	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺‘𝑥) ++ (𝐺‘𝑦))) = ((𝐹‘(𝐺‘𝑥)) ++ (𝐹‘(𝐺‘𝑦))))
46	37, 45	eqtrd 2771	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = ((𝐹‘(𝐺‘𝑥)) ++ (𝐹‘(𝐺‘𝑦))))
47	18, 22	syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
48	47	adantr 480	. . . . . . . 8 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
49	40, 48	eleqtrd 2838	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
50	42, 48	eleqtrd 2838	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51		ccatcl 14536	. . . . . . 7 ⊢ ((𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
52	49, 50, 51	syl2anc 585	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
53	52, 48	eleqtrrd 2839	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54		fvco3 6939	. . . . 5 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
55	39, 53, 54	syl2anc 585	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
56		fvco3 6939	. . . . . 6 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
57	39, 40, 56	syl2anc 585	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
58		fvco3 6939	. . . . . 6 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
59	39, 42, 58	syl2anc 585	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
60	57, 59	oveq12d 7385	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹 ∘ 𝐺)‘𝑥) ++ ((𝐹 ∘ 𝐺)‘𝑦)) = ((𝐹‘(𝐺‘𝑥)) ++ (𝐹‘(𝐺‘𝑦))))
61	46, 55, 60	3eqtr4d 2781	. . 3 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥) ++ ((𝐹 ∘ 𝐺)‘𝑦)))
62	61	ralrimivva 3180	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥) ++ ((𝐹 ∘ 𝐺)‘𝑦)))
63	1, 2, 21, 20	elmrsubrn 35702	. . 3 ⊢ (𝑇 ∈ V → ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥) ++ ((𝐹 ∘ 𝐺)‘𝑦)))))
64	18, 63	syl 17	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹 ∘ 𝐺)‘𝑥) ++ ((𝐹 ∘ 𝐺)‘𝑦)))))
65	8, 33, 62, 64	mpbir3and 1344	1 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887  ∅c0 4273  ran crn 5632   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Word cword 14475   ++ cconcat 14532  ⟨“cs1 14558  mCNcmcn 35642  mVRcmvar 35643  mRExcmrex 35648  mRSubstcmrsub 35652

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-gsum 17405  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-frmd 18817  df-vrmd 18818  df-mrex 35668  df-mrsub 35672

This theorem is referenced by:  msubco  35713