Theorem mrsubco 35722

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 2737	. . . . 5 ⊢ (mREx‘𝑇) = (mREx‘𝑇)
3	1, 2	mrsubf 35718	. . . 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 35718	. . . 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 6687	. . 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 4072	. . . . . . . . 9 ⊢ (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11		elun1 4123	. . . . . . . . 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 14560	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15		n0i 4281	. . . . . . . . . 10 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
16	1	rnfvprc 6829	. . . . . . . . . 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 2737	. . . . . . . 8 ⊢ (mCN‘𝑇) = (mCN‘𝑇)
21		eqid 2737	. . . . . . . 8 ⊢ (mVR‘𝑇) = (mVR‘𝑇)
22	20, 21, 2	mrexval 35702	. . . . . . 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 2840	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25		fvco3 6934	. . . . 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 35720	. . . . . 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 6839	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
30	1, 2, 21, 20	mrsubcn 35720	. . . . 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 2776	. . 3 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
33	32	ralrimiva 3130	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
34	1, 2	mrsubccat 35719	. . . . . . . 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 6839	. . . . 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 7032	. . . . . 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 7032	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘𝑦) ∈ (mREx‘𝑇))
44	1, 2	mrsubccat 35719	. . . . . 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 2772	. . . 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 2839	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
50	42, 48	eleqtrd 2839	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51		ccatcl 14530	. . . . . . 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 2840	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54		fvco3 6934	. . . . 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 6934	. . . . . 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 6934	. . . . . 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 7379	. . . 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 2782	. . 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 3181	. 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 35721	. . 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 3052  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888  ∅c0 4274  ran crn 5626   ∘ ccom 5629  ⟶wf 6489  ‘cfv 6493  (class class class)co 7361  Word cword 14469   ++ cconcat 14526  ⟨“cs1 14552  mCNcmcn 35661  mVRcmvar 35662  mRExcmrex 35667  mRSubstcmrsub 35671

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-n0 12432  df-xnn0 12505  df-z 12519  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-word 14470  df-lsw 14519  df-concat 14527  df-s1 14553  df-substr 14598  df-pfx 14628  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-0g 17398  df-gsum 17399  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-frmd 18811  df-vrmd 18812  df-mrex 35687  df-mrsub 35691

This theorem is referenced by:  msubco  35732