Theorem mplmon2mul 22014

Description: Product of scaled monomials. (Contributed by Stefan O'Rear, 8-Mar-2015.)

Hypotheses

Ref	Expression
mplmon2cl.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplmon2cl.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
mplmon2cl.z	⊢ 0 = (0g‘𝑅)
mplmon2cl.c	⊢ 𝐶 = (Base‘𝑅)
mplmon2cl.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplmon2mul.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mplmon2mul.t	⊢ ∙ = (.r‘𝑃)
mplmon2mul.u	⊢ · = (.r‘𝑅)
mplmon2mul.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
mplmon2mul.y	⊢ (𝜑 → 𝑌 ∈ 𝐷)
mplmon2mul.f	⊢ (𝜑 → 𝐹 ∈ 𝐶)
mplmon2mul.g	⊢ (𝜑 → 𝐺 ∈ 𝐶)

Assertion

Ref	Expression
mplmon2mul	⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (𝐹 · 𝐺), 0 )))

Distinct variable groups:   𝜑,𝑦   𝑦,𝐶   𝑦,𝐷   𝑦,𝐹   𝑦,𝐺   𝑓,𝐼   𝑦,𝑅   𝑦, ·   𝑓,𝑋,𝑦   𝑓,𝑌,𝑦   𝑦, 0

Allowed substitution hints:   𝜑(𝑓)   𝐶(𝑓)   𝐷(𝑓)   𝑃(𝑦,𝑓)   𝑅(𝑓)   ∙ (𝑦,𝑓)   · (𝑓)   𝐹(𝑓)   𝐺(𝑓)   𝐼(𝑦)   𝑊(𝑦,𝑓)   0 (𝑓)

Proof of Theorem mplmon2mul

Step	Hyp	Ref	Expression
1		mplmon2cl.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
2		mplmon2mul.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
3		mplmon2cl.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
4	3	mplassa 21969	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
5	1, 2, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
6		mplmon2mul.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐶)
7		mplmon2cl.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑅)
8	3, 1, 2	mplsca 21960	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
9	8	fveq2d 6835	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
10	7, 9	eqtrid 2780	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘(Scalar‘𝑃)))
11	6, 10	eleqtrd 2835	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑃)))
12		eqid 2733	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
13		mplmon2cl.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
14		eqid 2733	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15		mplmon2cl.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
16		crngring 20173	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
17	2, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
18		mplmon2mul.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
19	3, 12, 13, 14, 15, 1, 17, 18	mplmon 21980	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
20		assalmod 21807	. . . . . 6 ⊢ (𝑃 ∈ AssAlg → 𝑃 ∈ LMod)
21	5, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ LMod)
22		mplmon2mul.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
23	22, 10	eleqtrd 2835	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘(Scalar‘𝑃)))
24		mplmon2mul.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
25	3, 12, 13, 14, 15, 1, 17, 24	mplmon 21980	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
26		eqid 2733	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
27		eqid 2733	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
28		eqid 2733	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	12, 26, 27, 28	lmodvscl 20821	. . . . 5 ⊢ ((𝑃 ∈ LMod ∧ 𝐺 ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃)) → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) ∈ (Base‘𝑃))
30	21, 23, 25, 29	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) ∈ (Base‘𝑃))
31		mplmon2mul.t	. . . . 5 ⊢ ∙ = (.r‘𝑃)
32	12, 26, 28, 27, 31	assaass 21805	. . . 4 ⊢ ((𝑃 ∈ AssAlg ∧ (𝐹 ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃) ∧ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) ∈ (Base‘𝑃))) → ((𝐹( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 ))) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = (𝐹( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))))
33	5, 11, 19, 30, 32	syl13anc 1374	. . 3 ⊢ (𝜑 → ((𝐹( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 ))) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = (𝐹( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))))
34	12, 26, 28, 27, 31	assaassr 21806	. . . . 5 ⊢ ((𝑃 ∈ AssAlg ∧ (𝐺 ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))) → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = (𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))))
35	5, 23, 19, 25, 34	syl13anc 1374	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = (𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))))
36	35	oveq2d 7371	. . 3 ⊢ (𝜑 → (𝐹( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))) = (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))))
37	3, 12, 13, 14, 15, 1, 17, 18, 31, 24	mplmonmul 21981	. . . . . 6 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )))
38	37	oveq2d 7371	. . . . 5 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))))
39	38	oveq2d 7371	. . . 4 ⊢ (𝜑 → (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))) = (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )))))
40	15	psrbagaddcl 21871	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋 ∘f + 𝑌) ∈ 𝐷)
41	18, 24, 40	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ 𝐷)
42	3, 12, 13, 14, 15, 1, 17, 41	mplmon 21980	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
43		eqid 2733	. . . . . 6 ⊢ (.r‘(Scalar‘𝑃)) = (.r‘(Scalar‘𝑃))
44	12, 26, 27, 28, 43	lmodvsass 20830	. . . . 5 ⊢ ((𝑃 ∈ LMod ∧ (𝐹 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝐺 ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )) ∈ (Base‘𝑃))) → ((𝐹(.r‘(Scalar‘𝑃))𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))) = (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )))))
45	21, 11, 23, 42, 44	syl13anc 1374	. . . 4 ⊢ (𝜑 → ((𝐹(.r‘(Scalar‘𝑃))𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))) = (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )))))
46		mplmon2mul.u	. . . . . . 7 ⊢ · = (.r‘𝑅)
47	8	fveq2d 6835	. . . . . . 7 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(Scalar‘𝑃)))
48	46, 47	eqtr2id 2781	. . . . . 6 ⊢ (𝜑 → (.r‘(Scalar‘𝑃)) = · )
49	48	oveqd 7372	. . . . 5 ⊢ (𝜑 → (𝐹(.r‘(Scalar‘𝑃))𝐺) = (𝐹 · 𝐺))
50	49	oveq1d 7370	. . . 4 ⊢ (𝜑 → ((𝐹(.r‘(Scalar‘𝑃))𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))) = ((𝐹 · 𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))))
51	39, 45, 50	3eqtr2d 2774	. . 3 ⊢ (𝜑 → (𝐹( ·𝑠 ‘𝑃)(𝐺( ·𝑠 ‘𝑃)((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))))) = ((𝐹 · 𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))))
52	33, 36, 51	3eqtrd 2772	. 2 ⊢ (𝜑 → ((𝐹( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 ))) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = ((𝐹 · 𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))))
53	3, 27, 15, 14, 13, 7, 1, 17, 18, 6	mplmon2 22006	. . 3 ⊢ (𝜑 → (𝐹( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )))
54	3, 27, 15, 14, 13, 7, 1, 17, 24, 22	mplmon2 22006	. . 3 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 )))
55	53, 54	oveq12d 7373	. 2 ⊢ (𝜑 → ((𝐹( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 ))) ∙ (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )))) = ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))))
56	7, 46	ringcl 20178	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ∧ 𝐺 ∈ 𝐶) → (𝐹 · 𝐺) ∈ 𝐶)
57	17, 6, 22, 56	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐶)
58	3, 27, 15, 14, 13, 7, 1, 17, 41, 57	mplmon2 22006	. 2 ⊢ (𝜑 → ((𝐹 · 𝐺)( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (𝐹 · 𝐺), 0 )))
59	52, 55, 58	3eqtr3d 2776	1 ⊢ (𝜑 → ((𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, 𝐹, 0 )) ∙ (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (𝐹 · 𝐺), 0 )))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  {crab 3397  ifcif 4476   ↦ cmpt 5176  ^◡ccnv 5620   “ cima 5624  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617   ↑_m cmap 8759  Fincfn 8878   + caddc 11019  ℕcn 12135  ℕ₀cn0 12391  Basecbs 17130  ._rcmulr 17172  Scalarcsca 17174   ·_𝑠 cvsca 17175  0_gc0g 17353  1_rcur 20109  Ringcrg 20161  CRingccrg 20162  LModclmod 20803  AssAlgcasa 21797   mPoly cmpl 21853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11072  ax-resscn 11073  ax-1cn 11074  ax-icn 11075  ax-addcl 11076  ax-addrcl 11077  ax-mulcl 11078  ax-mulrcl 11079  ax-mulcom 11080  ax-addass 11081  ax-mulass 11082  ax-distr 11083  ax-i2m1 11084  ax-1ne0 11085  ax-1rid 11086  ax-rnegex 11087  ax-rrecex 11088  ax-cnre 11089  ax-pre-lttri 11090  ax-pre-lttrn 11091  ax-pre-ltadd 11092  ax-pre-mulgt0 11093

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619  df-ofr 7620  df-om 7806  df-1st 7930  df-2nd 7931  df-supp 8100  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8631  df-map 8761  df-pm 8762  df-ixp 8831  df-en 8879  df-dom 8880  df-sdom 8881  df-fin 8882  df-fsupp 9256  df-sup 9336  df-oi 9406  df-card 9842  df-pnf 11158  df-mnf 11159  df-xr 11160  df-ltxr 11161  df-le 11162  df-sub 11356  df-neg 11357  df-nn 12136  df-2 12198  df-3 12199  df-4 12200  df-5 12201  df-6 12202  df-7 12203  df-8 12204  df-9 12205  df-n0 12392  df-z 12479  df-dec 12599  df-uz 12743  df-fz 13418  df-fzo 13565  df-seq 13919  df-hash 14248  df-struct 17068  df-sets 17085  df-slot 17103  df-ndx 17115  df-base 17131  df-ress 17152  df-plusg 17184  df-mulr 17185  df-sca 17187  df-vsca 17188  df-ip 17189  df-tset 17190  df-ple 17191  df-ds 17193  df-hom 17195  df-cco 17196  df-0g 17355  df-gsum 17356  df-prds 17361  df-pws 17363  df-mre 17498  df-mrc 17499  df-acs 17501  df-mgm 18558  df-sgrp 18637  df-mnd 18653  df-mhm 18701  df-submnd 18702  df-grp 18859  df-minusg 18860  df-sbg 18861  df-mulg 18991  df-subg 19046  df-ghm 19135  df-cntz 19239  df-cmn 19704  df-abl 19705  df-mgp 20069  df-rng 20081  df-ur 20110  df-ring 20163  df-cring 20164  df-subrng 20471  df-subrg 20495  df-lmod 20805  df-lss 20875  df-assa 21800  df-psr 21856  df-mpl 21858

This theorem is referenced by:  evlslem2  22024