Theorem mplasclco 34013

Description: Case where composing an algebra scalar lifting functions with a scalar leads to a scalar. This is useful when working with selectVars. (Contributed by Thierry Arnoux, 4-May-2026.)

Hypotheses

Ref	Expression
mplasclco.s	⊢ 𝑆 = (Base‘𝑅)
mplasclco.o	⊢ 𝑂 = (𝐽 mPoly 𝑅)
mplasclco.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplasclco.q	⊢ 𝑄 = (𝐼 mPoly 𝑂)
mplasclco.a	⊢ 𝐴 = (algSc‘𝑂)
mplasclco.b	⊢ 𝐵 = (algSc‘𝑃)
mplasclco.c	⊢ 𝐶 = (algSc‘𝑄)
mplasclco.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
mplasclco.e	⊢ 𝐸 = {𝑗 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑗 “ ℕ) ∈ Fin}
mplasclco.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mplasclco.j	⊢ (𝜑 → 𝐽 ⊆ 𝐼)
mplasclco.r	⊢ (𝜑 → 𝑅 ∈ CRing)
mplasclco.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
mplasclco	⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

Distinct variable groups:   ℎ,𝐼   𝑗,𝐽   ℎ,𝑂   𝑅,ℎ   𝑅,𝑗

Allowed substitution hints:   𝜑(ℎ, 𝑗)   𝐴(ℎ, 𝑗)   𝐵(ℎ, 𝑗)   𝐶(ℎ, 𝑗)   𝐷(ℎ, 𝑗)   𝑃(ℎ, 𝑗)   𝑄(ℎ, 𝑗)   𝑆(ℎ, 𝑗)   𝐸(ℎ, 𝑗)   𝐼(𝑗)   𝐽(ℎ)   𝑂(𝑗)   𝑉(ℎ, 𝑗)   𝑋(ℎ, 𝑗)

Proof of Theorem mplasclco

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplasclco.o	. . . . 5 ⊢ 𝑂 = (𝐽 mPoly 𝑅)
2		eqid 2762	. . . . 5 ⊢ (Base‘𝑂) = (Base‘𝑂)
3		mplasclco.s	. . . . 5 ⊢ 𝑆 = (Base‘𝑅)
4		mplasclco.a	. . . . 5 ⊢ 𝐴 = (algSc‘𝑂)
5		mplasclco.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
6		mplasclco.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
7	5, 6	ssexd 5293	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8		mplasclco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
9	8	crngringd 20386	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 2, 3, 4, 7, 9	mplasclf 22282	. . . 4 ⊢ (𝜑 → 𝐴:𝑆⟶(Base‘𝑂))
11		mplasclco.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
12		mplasclco.d	. . . . . 6 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
13		eqid 2762	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
14		mplasclco.b	. . . . . 6 ⊢ 𝐵 = (algSc‘𝑃)
15		mplasclco.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
16	11, 12, 13, 3, 14, 5, 9, 15	mplascl 22281	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑋) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅))))
17	8	crnggrpd 20387	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	3, 13, 17	grpidcld 33466	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝑆)
19	15, 18	ifcld 4532	. . . . . 6 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
21	16, 20	fmpt3d 7112	. . . 4 ⊢ (𝜑 → (𝐵‘𝑋):𝐷⟶𝑆)
22	10, 21	fcod 6732	. . 3 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)):𝐷⟶(Base‘𝑂))
23	22	ffnd 6707	. 2 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) Fn 𝐷)
24		mplasclco.q	. . . . 5 ⊢ 𝑄 = (𝐼 mPoly 𝑂)
25		eqid 2762	. . . . 5 ⊢ (0g‘𝑂) = (0g‘𝑂)
26		mplasclco.c	. . . . 5 ⊢ 𝐶 = (algSc‘𝑄)
27	1, 7, 9	mplringd 22238	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
28		eqid 2762	. . . . . 6 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
29		eqid 2762	. . . . . 6 ⊢ (Base‘(Scalar‘𝑂)) = (Base‘(Scalar‘𝑂))
30	1	mplassa 22237	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝑅 ∈ CRing) → 𝑂 ∈ AssAlg)
31	7, 8, 30	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ AssAlg)
32	1, 7, 8	mplsca 22228	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
33	32	fveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑂)))
34	3, 33	eqtrid 2809	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑂)))
35	15, 34	eleqtrd 2864	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑂)))
36	4, 28, 29, 31, 35	asclelbas 22099	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘𝑂))
37	24, 12, 25, 2, 26, 5, 27, 36	mplascl 22281	. . . 4 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂))))
38	27	ringgrpd 20382	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Grp)
39	2, 25, 38	grpidcld 33466	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) ∈ (Base‘𝑂))
40	36, 39	ifcld 4532	. . . . 5 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
41	40	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
42	37, 41	fmpt3d 7112	. . 3 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)):𝐷⟶(Base‘𝑂))
43	42	ffnd 6707	. 2 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) Fn 𝐷)
44		eqeq2 2774	. . . . 5 ⊢ ((𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)})) → ((𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))) ↔ (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)}))))
45		eqeq2 2774	. . . . 5 ⊢ ((𝐸 × {(0g‘𝑅)}) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)})) → ((𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = (𝐸 × {(0g‘𝑅)}) ↔ (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)}))))
46		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑛 = (𝐼 × {0})) → 𝑛 = (𝐼 × {0}))
47	46	iftrued 4493	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑛 = (𝐼 × {0})) → if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅)) = if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)))
48	47	mpteq2dv 5203	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑛 = (𝐼 × {0})) → (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
49		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → ¬ 𝑛 = (𝐼 × {0}))
50	49	iffalsed 4496	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅)) = (0g‘𝑅))
51	50	mpteq2dv 5203	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = (𝑚 ∈ 𝐸 ↦ (0g‘𝑅)))
52		fconstmpt 5721	. . . . . 6 ⊢ (𝐸 × {(0g‘𝑅)}) = (𝑚 ∈ 𝐸 ↦ (0g‘𝑅))
53	51, 52	eqtr4di 2815	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = (𝐸 × {(0g‘𝑅)}))
54	44, 45, 48, 53	ifbothda 4524	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)})))
55		mplasclco.e	. . . . . 6 ⊢ 𝐸 = {𝑗 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑗 “ ℕ) ∈ Fin}
56	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝐽 ∈ V)
57	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑅 ∈ Ring)
58	21	ffvelcdmda 7080	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) ∈ 𝑆)
59	1, 55, 13, 3, 4, 56, 57, 58	mplascl 22281	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))))
60	16, 20	fvmpt2d 7004	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
61	60	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑚 ∈ 𝐸) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
62	61	ifeq1d 4505	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑚 ∈ 𝐸) → if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅)) = if(𝑚 = (𝐽 × {0}), if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅)))
63		ififcom 33011	. . . . . . 7 ⊢ if(𝑚 = (𝐽 × {0}), if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅)) = if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))
64	62, 63	eqtrdi 2813	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑚 ∈ 𝐸) → if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅)) = if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅)))
65	64	mpteq2dva 5202	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))) = (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))))
66	59, 65	eqtrd 2797	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑛 = (𝐼 × {0}), if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅)), (0g‘𝑅))))
67	1, 55, 13, 3, 4, 7, 9, 15	mplascl 22281	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
68	1, 55, 13, 25, 7, 17	mpl0 22221	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) = (𝐸 × {(0g‘𝑅)}))
69	67, 68	ifeq12d 4507	. . . . 5 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)})))
70	69	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) = if(𝑛 = (𝐼 × {0}), (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))), (𝐸 × {(0g‘𝑅)})))
71	54, 66, 70	3eqtr4d 2807	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
72	21	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐵‘𝑋):𝐷⟶𝑆)
73		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑛 ∈ 𝐷)
74	72, 73	fvco3d 6983	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = (𝐴‘((𝐵‘𝑋)‘𝑛)))
75	37, 41	fvmpt2d 7004	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐶‘(𝐴‘𝑋))‘𝑛) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
76	71, 74, 75	3eqtr4d 2807	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = ((𝐶‘(𝐴‘𝑋))‘𝑛))
77	23, 43, 76	eqfnfvd 7029	1 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ⊆ wss 3902  ifcif 4485  {csn 4587   ↦ cmpt 5190   × cxp 5657  ^◡ccnv 5658   “ cima 5662   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  Fincfn 8955  0cc0 11127  ℕcn 12260  ℕ₀cn0 12531  Basecbs 17305  Scalarcsca 17349  0_gc0g 17528  Ringcrg 20373  CRingccrg 20374  AssAlgcasa 22066  algSccascl 22068   mPoly cmpl 22122

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-ofr 7682  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-pm 8832  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-sup 9415  df-oi 9485  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-fzo 13712  df-seq 14068  df-hash 14397  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-sca 17362  df-vsca 17363  df-ip 17364  df-tset 17365  df-ple 17366  df-ds 17368  df-hom 17370  df-cco 17371  df-0g 17530  df-gsum 17531  df-prds 17536  df-pws 17538  df-mre 17674  df-mrc 17675  df-acs 17677  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-submnd 18893  df-grp 19061  df-minusg 19062  df-sbg 19063  df-mulg 19192  df-subg 19247  df-ghm 19342  df-cntz 19445  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-cring 20376  df-subrng 20709  df-subrg 20733  df-lmod 21047  df-lss 21117  df-assa 22069  df-ascl 22071  df-psr 22125  df-mpl 22127

This theorem is used by:  selvascl  34014