Theorem mplmon2mul 22129

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 22080	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
5	1, 2, 4	syl2anc 593	. . . 4 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
6		mplmon2mul.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐶)
7		mplmon2cl.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑅)
8	3, 1, 2	mplsca 22071	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
9	8	fveq2d 6871	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
10	7, 9	eqtrid 2810	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘(Scalar‘𝑃)))
11	6, 10	eleqtrd 2865	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑃)))
12		eqid 2763	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
13		mplmon2cl.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
14		eqid 2763	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15		mplmon2cl.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
16		crngring 20305	. . . . . 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 22095	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
20		assalmod 21919	. . . . . 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 2865	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘(Scalar‘𝑃)))
24		mplmon2mul.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
25	3, 12, 13, 14, 15, 1, 17, 24	mplmon 22095	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
26		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
27		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
28		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	12, 26, 27, 28	lmodvscl 20952	. . . . 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 1392	. . . 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 21917	. . . 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 1393	. . 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 21918	. . . . 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 1393	. . . 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 7412	. . 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 22096	. . . . . 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 7412	. . . . 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 7412	. . . 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 21983	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋 ∘f + 𝑌) ∈ 𝐷)
41	18, 24, 40	syl2anc 593	. . . . . 6 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ 𝐷)
42	3, 12, 13, 14, 15, 1, 17, 41	mplmon 22095	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
43		eqid 2763	. . . . . 6 ⊢ (.r‘(Scalar‘𝑃)) = (.r‘(Scalar‘𝑃))
44	12, 26, 27, 28, 43	lmodvsass 20961	. . . . 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 1393	. . . 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 6871	. . . . . . 7 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(Scalar‘𝑃)))
48	46, 47	eqtr2id 2811	. . . . . 6 ⊢ (𝜑 → (.r‘(Scalar‘𝑃)) = · )
49	48	oveqd 7413	. . . . 5 ⊢ (𝜑 → (𝐹(.r‘(Scalar‘𝑃))𝐺) = (𝐹 · 𝐺))
50	49	oveq1d 7411	. . . 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 2804	. . 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 2802	. 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 22121	. . 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 22121	. . 3 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 )))
55	53, 54	oveq12d 7414	. 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 20310	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ∧ 𝐺 ∈ 𝐶) → (𝐹 · 𝐺) ∈ 𝐶)
57	17, 6, 22, 56	syl3anc 1392	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐶)
58	3, 27, 15, 14, 13, 7, 1, 17, 41, 57	mplmon2 22121	. 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 2806	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 1561   ∈ wcel 2143  {crab 3415  ifcif 4481   ↦ cmpt 5182  ^◡ccnv 5647   “ cima 5651  ‘cfv 6521  (class class class)co 7396   ∘_f cof 7658   ↑_m cmap 8808  Fincfn 8927   + caddc 11087  ℕcn 12220  ℕ₀cn0 12491  Basecbs 17255  ._rcmulr 17297  Scalarcsca 17299   ·_𝑠 cvsca 17300  0_gc0g 17478  1_rcur 20241  Ringcrg 20293  CRingccrg 20294  LModclmod 20934  AssAlgcasa 21909   mPoly cmpl 21965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-sup 9386  df-oi 9456  df-card 9909  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-fzo 13670  df-seq 14025  df-hash 14354  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-sca 17312  df-vsca 17313  df-ip 17314  df-tset 17315  df-ple 17316  df-ds 17318  df-hom 17320  df-cco 17321  df-0g 17480  df-gsum 17481  df-prds 17486  df-pws 17488  df-mre 17624  df-mrc 17625  df-acs 17627  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-submnd 18828  df-grp 18988  df-minusg 18989  df-sbg 18990  df-mulg 19120  df-subg 19175  df-ghm 19264  df-cntz 19367  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-ring 20295  df-cring 20296  df-subrng 20606  df-subrg 20630  df-lmod 20936  df-lss 21006  df-assa 21912  df-psr 21968  df-mpl 21970

This theorem is referenced by:  evlslem2  22139