Theorem mrsubco 33150

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 33146	. . . 4 ⊢ (𝐹 ∈ ran 𝑆 → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
4	3	adantr 484	. . 3 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
5	1, 2	mrsubf 33146	. . . 4 ⊢ (𝐺 ∈ ran 𝑆 → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
6	5	adantl 485	. . 3 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7		fco 6547	. . 3 ⊢ ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
8	4, 6, 7	syl2anc 587	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → (𝐹 ∘ 𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
9	6	adantr 484	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
10		eldifi 4027	. . . . . . . . 9 ⊢ (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11		elun1 4076	. . . . . . . . 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 485	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
14	13	s1cld 14125	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15		n0i 4234	. . . . . . . . . 10 ⊢ (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
16	1	rnfvprc 6689	. . . . . . . . . 10 ⊢ (¬ 𝑇 ∈ V → ran 𝑆 = ∅)
17	15, 16	nsyl2 143	. . . . . . . . 9 ⊢ (𝐹 ∈ ran 𝑆 → 𝑇 ∈ V)
18	17	adantr 484	. . . . . . . 8 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
19	18	adantr 484	. . . . . . 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 33130	. . . . . . 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 2834	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25		fvco3 6788	. . . . 5 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
26	9, 24, 25	syl2anc 587	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
27	1, 2, 21, 20	mrsubcn 33148	. . . . . 6 ⊢ ((𝐺 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
28	27	adantll 714	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
29	28	fveq2d 6699	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
30	1, 2, 21, 20	mrsubcn 33148	. . . . 5 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
31	30	adantlr 715	. . . 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 3095	. 2 ⊢ ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹 ∘ 𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
34	1, 2	mrsubccat 33147	. . . . . . . 8 ⊢ ((𝐺 ∈ ran 𝑆 ∧ 𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
35	34	3expb 1122	. . . . . . 7 ⊢ ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
36	35	adantll 714	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺‘𝑥) ++ (𝐺‘𝑦)))
37	36	fveq2d 6699	. . . . 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 484	. . . . . . 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	ffvelrnd 6883	. . . . . 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	ffvelrnd 6883	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘𝑦) ∈ (mREx‘𝑇))
44	1, 2	mrsubccat 33147	. . . . . 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 1373	. . . . 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 484	. . . . . . . 8 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
49	40, 48	eleqtrd 2833	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
50	42, 48	eleqtrd 2833	. . . . . . 7 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51		ccatcl 14094	. . . . . . 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 587	. . . . . 6 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
53	52, 48	eleqtrrd 2834	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54		fvco3 6788	. . . . 5 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
55	39, 53, 54	syl2anc 587	. . . 4 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
56		fvco3 6788	. . . . . 6 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
57	39, 40, 56	syl2anc 587	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥)))
58		fvco3 6788	. . . . . 6 ⊢ ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
59	39, 42, 58	syl2anc 587	. . . . 5 ⊢ (((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦)))
60	57, 59	oveq12d 7209	. . . 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 3102	. 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 33149	. . 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 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051  Vcvv 3398   ∖ cdif 3850   ∪ cun 3851  ∅c0 4223  ran crn 5537   ∘ ccom 5540  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Word cword 14034   ++ cconcat 14090  ⟨“cs1 14117  mCNcmcn 33089  mVRcmvar 33090  mRExcmrex 33095  mRSubstcmrsub 33099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-er 8369  df-map 8488  df-pm 8489  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-card 9520  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-n0 12056  df-xnn0 12128  df-z 12142  df-uz 12404  df-fz 13061  df-fzo 13204  df-seq 13540  df-hash 13862  df-word 14035  df-lsw 14083  df-concat 14091  df-s1 14118  df-substr 14171  df-pfx 14201  df-struct 16668  df-ndx 16669  df-slot 16670  df-base 16672  df-sets 16673  df-ress 16674  df-plusg 16762  df-0g 16900  df-gsum 16901  df-mgm 18068  df-sgrp 18117  df-mnd 18128  df-mhm 18172  df-submnd 18173  df-frmd 18230  df-vrmd 18231  df-mrex 33115  df-mrsub 33119

This theorem is referenced by:  msubco  33160