Theorem mplmon2mul 21187

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 21137	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
5	1, 2, 4	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
6		mplmon2mul.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐶)
7		mplmon2cl.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑅)
8	3, 1, 2	mplsca 21127	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
9	8	fveq2d 6760	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
10	7, 9	eqtrid 2790	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘(Scalar‘𝑃)))
11	6, 10	eleqtrd 2841	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑃)))
12		eqid 2738	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
13		mplmon2cl.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
14		eqid 2738	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15		mplmon2cl.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
16		crngring 19710	. . . . . 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 21146	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
20		assalmod 20977	. . . . . 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 2841	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘(Scalar‘𝑃)))
24		mplmon2mul.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
25	3, 12, 13, 14, 15, 1, 17, 24	mplmon 21146	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
26		eqid 2738	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
27		eqid 2738	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
28		eqid 2738	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	12, 26, 27, 28	lmodvscl 20055	. . . . 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 1369	. . . 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 20975	. . . 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 1370	. . 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 20976	. . . . 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 1370	. . . 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 7271	. . 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 21147	. . . . . 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 7271	. . . . 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 7271	. . . 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 21041	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋 ∘f + 𝑌) ∈ 𝐷)
41	18, 24, 40	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ 𝐷)
42	3, 12, 13, 14, 15, 1, 17, 41	mplmon 21146	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
43		eqid 2738	. . . . . 6 ⊢ (.r‘(Scalar‘𝑃)) = (.r‘(Scalar‘𝑃))
44	12, 26, 27, 28, 43	lmodvsass 20063	. . . . 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 1370	. . . 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 6760	. . . . . . 7 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(Scalar‘𝑃)))
48	46, 47	eqtr2id 2792	. . . . . 6 ⊢ (𝜑 → (.r‘(Scalar‘𝑃)) = · )
49	48	oveqd 7272	. . . . 5 ⊢ (𝜑 → (𝐹(.r‘(Scalar‘𝑃))𝐺) = (𝐹 · 𝐺))
50	49	oveq1d 7270	. . . 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 2784	. . 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 2782	. 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 21179	. . 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 21179	. . 3 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 )))
55	53, 54	oveq12d 7273	. 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 19715	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ∧ 𝐺 ∈ 𝐶) → (𝐹 · 𝐺) ∈ 𝐶)
57	17, 6, 22, 56	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐶)
58	3, 27, 15, 14, 13, 7, 1, 17, 41, 57	mplmon2 21179	. 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 2786	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 1539   ∈ wcel 2108  {crab 3067  ifcif 4456   ↦ cmpt 5153  ^◡ccnv 5579   “ cima 5583  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509   ↑_m cmap 8573  Fincfn 8691   + caddc 10805  ℕcn 11903  ℕ₀cn0 12163  Basecbs 16840  ._rcmulr 16889  Scalarcsca 16891   ·_𝑠 cvsca 16892  0_gc0g 17067  1_rcur 19652  Ringcrg 19698  CRingccrg 19699  LModclmod 20038  AssAlgcasa 20967   mPoly cmpl 21019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-ofr 7512  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-seq 13650  df-hash 13973  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-tset 16907  df-0g 17069  df-gsum 17070  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-mhm 18345  df-submnd 18346  df-grp 18495  df-minusg 18496  df-sbg 18497  df-mulg 18616  df-subg 18667  df-ghm 18747  df-cntz 18838  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-cring 19701  df-subrg 19937  df-lmod 20040  df-lss 20109  df-assa 20970  df-psr 21022  df-mpl 21024

This theorem is referenced by:  evlslem2  21199