Theorem mplmon2mul 22408

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 22359	. . . . 5 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ AssAlg)
5	1, 2, 4	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝑃 ∈ AssAlg)
6		mplmon2mul.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐶)
7		mplmon2cl.c	. . . . . 6 ⊢ 𝐶 = (Base‘𝑅)
8	3, 1, 2	mplsca 22350	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
9	8	fveq2d 6912	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
10	7, 9	eqtrid 2810	. . . . 5 ⊢ (𝜑 → 𝐶 = (Base‘(Scalar‘𝑃)))
11	6, 10	eleqtrd 2866	. . . 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 20502	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
17	2, 16	syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
18		mplmon2mul.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
19	3, 12, 13, 14, 15, 1, 17, 18	mplmon 22374	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑋, (1r‘𝑅), 0 )) ∈ (Base‘𝑃))
20		assalmod 22198	. . . . . 6 ⊢ (𝑃 ∈ AssAlg → 𝑃 ∈ LMod)
21	5, 20	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ LMod)
22		mplmon2mul.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
23	22, 10	eleqtrd 2866	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (Base‘(Scalar‘𝑃)))
24		mplmon2mul.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐷)
25	3, 12, 13, 14, 15, 1, 17, 24	mplmon 22374	. . . . 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 21183	. . . . 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 1399	. . . 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 22196	. . . 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 1400	. . 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 22197	. . . . 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 1400	. . . 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 7459	. . 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 22375	. . . . . 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 7459	. . . . 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 7459	. . . 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 22262	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑌 ∈ 𝐷) → (𝑋 ∘f + 𝑌) ∈ 𝐷)
41	18, 24, 40	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑋 ∘f + 𝑌) ∈ 𝐷)
42	3, 12, 13, 14, 15, 1, 17, 41	mplmon 22374	. . . . 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 21192	. . . . 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 1400	. . . 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 6912	. . . . . . 7 ⊢ (𝜑 → (.r‘𝑅) = (.r‘(Scalar‘𝑃)))
48	46, 47	eqtr2id 2811	. . . . . 6 ⊢ (𝜑 → (.r‘(Scalar‘𝑃)) = · )
49	48	oveqd 7460	. . . . 5 ⊢ (𝜑 → (𝐹(.r‘(Scalar‘𝑃))𝐺) = (𝐹 · 𝐺))
50	49	oveq1d 7458	. . . 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 22400	. . 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 22400	. . 3 ⊢ (𝜑 → (𝐺( ·𝑠 ‘𝑃)(𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, (1r‘𝑅), 0 ))) = (𝑦 ∈ 𝐷 ↦ if(𝑦 = 𝑌, 𝐺, 0 )))
55	53, 54	oveq12d 7461	. 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 20507	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐶 ∧ 𝐺 ∈ 𝐶) → (𝐹 · 𝐺) ∈ 𝐶)
57	17, 6, 22, 56	syl3anc 1399	. . 3 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐶)
58	3, 27, 15, 14, 13, 7, 1, 17, 41, 57	mplmon2 22400	. 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 )))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  ifcif 4485   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715   ↑_m cmap 8875  Fincfn 9001   + caddc 11230  ℕcn 12362  ℕ₀cn0 12633  Basecbs 17416  ._rcmulr 17458  Scalarcsca 17460   ·_𝑠 cvsca 17461  0_gc0g 17639  1_rcur 20437  Ringcrg 20489  CRingccrg 20490  LModclmod 21165  AssAlgcasa 22188   mPoly cmpl 22244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-cring 20492  df-subrng 20828  df-subrg 20852  df-lmod 21167  df-lss 21237  df-assa 22191  df-psr 22247  df-mpl 22249

This theorem is used by:  evlslem2  22418