Theorem mplasclco 34218

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 2763	. . . . 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 5290	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8		mplasclco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
9	8	crngringd 20503	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 2, 3, 4, 7, 9	mplasclf 22404	. . . 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 2763	. . . . . 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 22403	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑋) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅))))
17	8	crnggrpd 20504	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	3, 13, 17	grpidcld 33670	. . . . . . 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 7140	. . . 4 ⊢ (𝜑 → (𝐵‘𝑋):𝐷⟶𝑆)
22	10, 21	fcod 6757	. . 3 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)):𝐷⟶(Base‘𝑂))
23	22	ffnd 6732	. 2 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) Fn 𝐷)
24		mplasclco.q	. . . . 5 ⊢ 𝑄 = (𝐼 mPoly 𝑂)
25		eqid 2763	. . . . 5 ⊢ (0g‘𝑂) = (0g‘𝑂)
26		mplasclco.c	. . . . 5 ⊢ 𝐶 = (algSc‘𝑄)
27	1, 7, 9	mplringd 22360	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
28		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
29		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑂)) = (Base‘(Scalar‘𝑂))
30	1	mplassa 22359	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝑅 ∈ CRing) → 𝑂 ∈ AssAlg)
31	7, 8, 30	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ AssAlg)
32	1, 7, 8	mplsca 22350	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
33	32	fveq2d 6912	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑂)))
34	3, 33	eqtrid 2810	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑂)))
35	15, 34	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑂)))
36	4, 28, 29, 31, 35	asclelbas 22221	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘𝑂))
37	24, 12, 25, 2, 26, 5, 27, 36	mplascl 22403	. . . 4 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂))))
38	27	ringgrpd 20499	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Grp)
39	2, 25, 38	grpidcld 33670	. . . . . 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 7140	. . 3 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)):𝐷⟶(Base‘𝑂))
43	42	ffnd 6732	. 2 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) Fn 𝐷)
44		eqeq2 2775	. . . . 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 2775	. . . . 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 5202	. . . . 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 5202	. . . . . 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 5717	. . . . . 6 ⊢ (𝐸 × {(0g‘𝑅)}) = (𝑚 ∈ 𝐸 ↦ (0g‘𝑅))
53	51, 52	eqtr4di 2816	. . . . 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 7108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) ∈ 𝑆)
59	1, 55, 13, 3, 4, 56, 57, 58	mplascl 22403	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))))
60	16, 20	fvmpt2d 7030	. . . . . . . . 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 33218	. . . . . . 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 2814	. . . . . 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 5201	. . . . 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 2798	. . . 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 22403	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
68	1, 55, 13, 25, 7, 17	mpl0 22343	. . . . . 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 2808	. . 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 7009	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = (𝐴‘((𝐵‘𝑋)‘𝑛)))
75	37, 41	fvmpt2d 7030	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐶‘(𝐴‘𝑋))‘𝑛) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
76	71, 74, 75	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = ((𝐶‘(𝐴‘𝑋))‘𝑛))
77	23, 43, 76	eqfnfvd 7055	1 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900  ifcif 4485  {csn 4588   ↦ cmpt 5189   × cxp 5653  ^◡ccnv 5654   “ cima 5658   ∘ ccom 5659  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875  Fincfn 9001  0cc0 11227  ℕcn 12362  ℕ₀cn0 12633  Basecbs 17416  Scalarcsca 17460  0_gc0g 17639  Ringcrg 20489  CRingccrg 20490  AssAlgcasa 22188  algSccascl 22190   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-ascl 22193  df-psr 22247  df-mpl 22249

This theorem is used by:  selvascl  34219