Theorem mplmon2mul 22093

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 22042	. . . . 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 22033	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
9	8	fveq2d 6910	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
10	7, 9	eqtrid 2789	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘(Scalar‘𝑃)))
11	6, 10	eleqtrd 2843	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(Scalar‘𝑃)))
12		eqid 2737	. . . . 5 ⊢ (Base‘𝑃) = (Base‘𝑃)
13		mplmon2cl.z	. . . . 5 ⊢ 0 = (0g‘𝑅)
14		eqid 2737	. . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅)
15		mplmon2cl.d	. . . . 5 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
16		crngring 20242	. . . . . 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 22053	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
20		assalmod 21880	. . . . . 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 2843	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘(Scalar‘𝑃)))
24		mplmon2mul.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
25	3, 12, 13, 14, 15, 1, 17, 24	mplmon 22053	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
26		eqid 2737	. . . . . 6 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
27		eqid 2737	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
28		eqid 2737	. . . . . 6 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	12, 26, 27, 28	lmodvscl 20876	. . . . 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 21878	. . . 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 21879	. . . . 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 7447	. . 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 22054	. . . . . 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 7447	. . . . 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 7447	. . . 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 21944	. . . . . . 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 22053	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = (𝑋 ∘f + 𝑌), (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
43		eqid 2737	. . . . . 6 ⊢ (.r‘(Scalar‘𝑃)) = (.r‘(Scalar‘𝑃))
44	12, 26, 27, 28, 43	lmodvsass 20885	. . . . 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 6910	. . . . . . 7 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(Scalar‘𝑃)))
48	46, 47	eqtr2id 2790	. . . . . 6 ⊢ (𝜑 → (.r‘(Scalar‘𝑃)) = · )
49	48	oveqd 7448	. . . . 5 ⊢ (𝜑 → (𝐹(.r‘(Scalar‘𝑃))𝐺) = (𝐹 · 𝐺))
50	49	oveq1d 7446	. . . 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 2783	. . 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 2781	. 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 22085	. . 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 22085	. . 3 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 )))
55	53, 54	oveq12d 7449	. 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 20247	. . . 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 22085	. 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 2785	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 1540   ∈ wcel 2108  {crab 3436  ifcif 4525   ↦ cmpt 5225  ^◡ccnv 5684   “ cima 5688  ‘cfv 6561  (class class class)co 7431   ∘_f cof 7695   ↑_m cmap 8866  Fincfn 8985   + caddc 11158  ℕcn 12266  ℕ₀cn0 12526  Basecbs 17247  ._rcmulr 17298  Scalarcsca 17300   ·_𝑠 cvsca 17301  0_gc0g 17484  1_rcur 20178  Ringcrg 20230  CRingccrg 20231  LModclmod 20858  AssAlgcasa 21870   mPoly cmpl 21926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-assa 21873  df-psr 21929  df-mpl 21931

This theorem is referenced by:  evlslem2  22103