Theorem mplasclco 33915

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 5294	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8		mplasclco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
9	8	crngringd 20334	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 2, 3, 4, 7, 9	mplasclf 22227	. . . 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 22226	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑋) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅))))
17	8	crnggrpd 20335	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	3, 13, 17	grpidcld 33368	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝑆)
19	15, 18	ifcld 4533	. . . . . 6 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
20	19	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
21	16, 20	fmpt3d 7111	. . . 4 ⊢ (𝜑 → (𝐵‘𝑋):𝐷⟶𝑆)
22	10, 21	fcod 6731	. . 3 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)):𝐷⟶(Base‘𝑂))
23	22	ffnd 6706	. 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 22183	. . . . 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 22182	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝑅 ∈ CRing) → 𝑂 ∈ AssAlg)
31	7, 8, 30	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ AssAlg)
32	1, 7, 8	mplsca 22173	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
33	32	fveq2d 6885	. . . . . . . 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 22044	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘𝑂))
37	24, 12, 25, 2, 26, 5, 27, 36	mplascl 22226	. . . 4 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂))))
38	27	ringgrpd 20330	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Grp)
39	2, 25, 38	grpidcld 33368	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) ∈ (Base‘𝑂))
40	36, 39	ifcld 4533	. . . . 5 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
41	40	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
42	37, 41	fmpt3d 7111	. . 3 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)):𝐷⟶(Base‘𝑂))
43	42	ffnd 6706	. 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 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑛 = (𝐼 × {0})) → 𝑛 = (𝐼 × {0}))
47	46	iftrued 4494	. . . . . 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 5204	. . . . 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 489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → ¬ 𝑛 = (𝐼 × {0}))
50	49	iffalsed 4497	. . . . . . 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 5204	. . . . . 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 5722	. . . . . 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 4525	. . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝐽 ∈ V)
57	9	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑅 ∈ Ring)
58	21	ffvelcdmda 7079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) ∈ 𝑆)
59	1, 55, 13, 3, 4, 56, 57, 58	mplascl 22226	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))))
60	16, 20	fvmpt2d 7003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
61	60	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑚 ∈ 𝐸) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
62	61	ifeq1d 4506	. . . . . . 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 32907	. . . . . . 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 5203	. . . . 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 22226	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
68	1, 55, 13, 25, 7, 17	mpl0 22166	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) = (𝐸 × {(0g‘𝑅)}))
69	67, 68	ifeq12d 4508	. . . . 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 485	. . . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐵‘𝑋):𝐷⟶𝑆)
73		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑛 ∈ 𝐷)
74	72, 73	fvco3d 6982	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = (𝐴‘((𝐵‘𝑋)‘𝑛)))
75	37, 41	fvmpt2d 7003	. . 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 7028	1 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415  Vcvv 3454   ⊆ wss 3904  ifcif 4486  {csn 4588   ↦ cmpt 5191   × cxp 5658  ^◡ccnv 5659   “ cima 5663   ∘ ccom 5664  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ↑_m cmap 8822  Fincfn 8941  0cc0 11106  ℕcn 12239  ℕ₀cn0 12510  Basecbs 17275  Scalarcsca 17319  0_gc0g 17498  Ringcrg 20321  CRingccrg 20322  AssAlgcasa 22011  algSccascl 22013   mPoly cmpl 22067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-ofr 7677  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8155  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-er 8692  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9932  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-fzo 13690  df-seq 14045  df-hash 14374  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-sca 17332  df-vsca 17333  df-ip 17334  df-tset 17335  df-ple 17336  df-ds 17338  df-hom 17340  df-cco 17341  df-0g 17500  df-gsum 17501  df-prds 17506  df-pws 17508  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-mhm 18847  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-mulg 19140  df-subg 19195  df-ghm 19290  df-cntz 19393  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-cring 20324  df-subrng 20656  df-subrg 20680  df-lmod 20994  df-lss 21064  df-assa 22014  df-ascl 22016  df-psr 22070  df-mpl 22072

This theorem is used by:  selvascl  33916