Theorem mplasclco 34081

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 2760	. . . . 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 5285	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8		mplasclco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
9	8	crngringd 20434	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 2, 3, 4, 7, 9	mplasclf 22335	. . . 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 2760	. . . . . 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 22334	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑋) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅))))
17	8	crnggrpd 20435	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	3, 13, 17	grpidcld 33533	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝑆)
19	15, 18	ifcld 4528	. . . . . 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 7104	. . . 4 ⊢ (𝜑 → (𝐵‘𝑋):𝐷⟶𝑆)
22	10, 21	fcod 6723	. . 3 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)):𝐷⟶(Base‘𝑂))
23	22	ffnd 6698	. 2 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) Fn 𝐷)
24		mplasclco.q	. . . . 5 ⊢ 𝑄 = (𝐼 mPoly 𝑂)
25		eqid 2760	. . . . 5 ⊢ (0g‘𝑂) = (0g‘𝑂)
26		mplasclco.c	. . . . 5 ⊢ 𝐶 = (algSc‘𝑄)
27	1, 7, 9	mplringd 22291	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
28		eqid 2760	. . . . . 6 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
29		eqid 2760	. . . . . 6 ⊢ (Base‘(Scalar‘𝑂)) = (Base‘(Scalar‘𝑂))
30	1	mplassa 22290	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝑅 ∈ CRing) → 𝑂 ∈ AssAlg)
31	7, 8, 30	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ AssAlg)
32	1, 7, 8	mplsca 22281	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
33	32	fveq2d 6877	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑂)))
34	3, 33	eqtrid 2807	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑂)))
35	15, 34	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑂)))
36	4, 28, 29, 31, 35	asclelbas 22152	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘𝑂))
37	24, 12, 25, 2, 26, 5, 27, 36	mplascl 22334	. . . 4 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂))))
38	27	ringgrpd 20430	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Grp)
39	2, 25, 38	grpidcld 33533	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) ∈ (Base‘𝑂))
40	36, 39	ifcld 4528	. . . . 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 7104	. . 3 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)):𝐷⟶(Base‘𝑂))
43	42	ffnd 6698	. 2 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) Fn 𝐷)
44		eqeq2 2772	. . . . 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 2772	. . . . 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 4489	. . . . . 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 5198	. . . . 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 4492	. . . . . . 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 5198	. . . . . 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 5709	. . . . . 6 ⊢ (𝐸 × {(0g‘𝑅)}) = (𝑚 ∈ 𝐸 ↦ (0g‘𝑅))
53	51, 52	eqtr4di 2813	. . . . 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 4520	. . . 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 7072	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) ∈ 𝑆)
59	1, 55, 13, 3, 4, 56, 57, 58	mplascl 22334	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))))
60	16, 20	fvmpt2d 6995	. . . . . . . . 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 4501	. . . . . . 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 33079	. . . . . . 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 2811	. . . . . 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 5197	. . . . 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 2795	. . . 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 22334	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
68	1, 55, 13, 25, 7, 17	mpl0 22274	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) = (𝐸 × {(0g‘𝑅)}))
69	67, 68	ifeq12d 4503	. . . . 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 2805	. . 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 6974	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = (𝐴‘((𝐵‘𝑋)‘𝑛)))
75	37, 41	fvmpt2d 6995	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐶‘(𝐴‘𝑋))‘𝑛) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
76	71, 74, 75	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = ((𝐶‘(𝐴‘𝑋))‘𝑛))
77	23, 43, 76	eqfnfvd 7020	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 3412  Vcvv 3450   ⊆ wss 3898  ifcif 4481  {csn 4583   ↦ cmpt 5185   × cxp 5645  ^◡ccnv 5646   “ cima 5650   ∘ ccom 5651  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408   ↑_m cmap 8825  Fincfn 8951  0cc0 11171  ℕcn 12304  ℕ₀cn0 12575  Basecbs 17348  Scalarcsca 17392  0_gc0g 17571  Ringcrg 20420  CRingccrg 20421  AssAlgcasa 22119  algSccascl 22121   mPoly cmpl 22175

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7676  df-ofr 7677  df-om 7861  df-1st 7984  df-2nd 7985  df-supp 8156  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-er 8695  df-map 8827  df-pm 8828  df-ixp 8904  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-fsupp 9332  df-sup 9412  df-oi 9482  df-card 9991  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-uz 12935  df-fz 13609  df-fzo 13757  df-seq 14113  df-hash 14442  df-struct 17286  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-ress 17370  df-plusg 17402  df-mulr 17403  df-sca 17405  df-vsca 17406  df-ip 17407  df-tset 17408  df-ple 17409  df-ds 17411  df-hom 17413  df-cco 17414  df-0g 17573  df-gsum 17574  df-prds 17579  df-pws 17581  df-mre 17717  df-mrc 17718  df-acs 17720  df-mgm 18777  df-sgrp 18869  df-mnd 18885  df-mhm 18939  df-submnd 18940  df-grp 19108  df-minusg 19109  df-sbg 19110  df-mulg 19239  df-subg 19294  df-ghm 19389  df-cntz 19492  df-cmn 19957  df-abl 19958  df-mgp 20322  df-rng 20336  df-ur 20369  df-ring 20422  df-cring 20423  df-subrng 20759  df-subrg 20783  df-lmod 21098  df-lss 21168  df-assa 22122  df-ascl 22124  df-psr 22178  df-mpl 22180

This theorem is used by:  selvascl  34082