Theorem mplasclco 33703

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 2736	. . . . 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 5255	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ V)
8		mplasclco.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
9	8	crngringd 20221	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
10	1, 2, 3, 4, 7, 9	mplasclf 22044	. . . 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 2736	. . . . . 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 22043	. . . . 5 ⊢ (𝜑 → (𝐵‘𝑋) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅))))
17	8	crnggrpd 20222	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Grp)
18	3, 13, 17	grpidcld 33122	. . . . . . 7 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝑆)
19	15, 18	ifcld 4504	. . . . . 6 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)) ∈ 𝑆)
21	16, 20	fmpt3d 7060	. . . 4 ⊢ (𝜑 → (𝐵‘𝑋):𝐷⟶𝑆)
22	10, 21	fcod 6683	. . 3 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)):𝐷⟶(Base‘𝑂))
23	22	ffnd 6659	. 2 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) Fn 𝐷)
24		mplasclco.q	. . . . 5 ⊢ 𝑄 = (𝐼 mPoly 𝑂)
25		eqid 2736	. . . . 5 ⊢ (0g‘𝑂) = (0g‘𝑂)
26		mplasclco.c	. . . . 5 ⊢ 𝐶 = (algSc‘𝑄)
27	1, 7, 9	mplringd 22000	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ Ring)
28		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑂) = (Scalar‘𝑂)
29		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑂)) = (Base‘(Scalar‘𝑂))
30	1	mplassa 21999	. . . . . . 7 ⊢ ((𝐽 ∈ V ∧ 𝑅 ∈ CRing) → 𝑂 ∈ AssAlg)
31	7, 8, 30	syl2anc 586	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ AssAlg)
32	1, 7, 8	mplsca 21990	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑂))
33	32	fveq2d 6834	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑂)))
34	3, 33	eqtrid 2783	. . . . . . 7 ⊢ (𝜑 → 𝑆 = (Base‘(Scalar‘𝑂)))
35	15, 34	eleqtrd 2838	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘(Scalar‘𝑂)))
36	4, 28, 29, 31, 35	asclelbas 21861	. . . . 5 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘𝑂))
37	24, 12, 25, 2, 26, 5, 27, 36	mplascl 22043	. . . 4 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) = (𝑛 ∈ 𝐷 ↦ if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂))))
38	27	ringgrpd 20217	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Grp)
39	2, 25, 38	grpidcld 33122	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) ∈ (Base‘𝑂))
40	36, 39	ifcld 4504	. . . . 5 ⊢ (𝜑 → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
41	40	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)) ∈ (Base‘𝑂))
42	37, 41	fmpt3d 7060	. . 3 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)):𝐷⟶(Base‘𝑂))
43	42	ffnd 6659	. 2 ⊢ (𝜑 → (𝐶‘(𝐴‘𝑋)) Fn 𝐷)
44		eqeq2 2748	. . . . 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 2748	. . . . 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 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑛 = (𝐼 × {0})) → 𝑛 = (𝐼 × {0}))
47	46	iftrued 4465	. . . . . 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 5169	. . . . 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 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ ¬ 𝑛 = (𝐼 × {0})) → ¬ 𝑛 = (𝐼 × {0}))
50	49	iffalsed 4468	. . . . . . 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 5169	. . . . . 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 5683	. . . . . 6 ⊢ (𝐸 × {(0g‘𝑅)}) = (𝑚 ∈ 𝐸 ↦ (0g‘𝑅))
53	51, 52	eqtr4di 2789	. . . . 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 4496	. . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝐽 ∈ V)
57	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑅 ∈ Ring)
58	21	ffvelcdmda 7028	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) ∈ 𝑆)
59	1, 55, 13, 3, 4, 56, 57, 58	mplascl 22043	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), ((𝐵‘𝑋)‘𝑛), (0g‘𝑅))))
60	16, 20	fvmpt2d 6952	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
61	60	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝐷) ∧ 𝑚 ∈ 𝐸) → ((𝐵‘𝑋)‘𝑛) = if(𝑛 = (𝐼 × {0}), 𝑋, (0g‘𝑅)))
62	61	ifeq1d 4477	. . . . . . 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 32641	. . . . . . 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 2787	. . . . . 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 5168	. . . . 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 2771	. . . 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 22043	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = (𝑚 ∈ 𝐸 ↦ if(𝑚 = (𝐽 × {0}), 𝑋, (0g‘𝑅))))
68	1, 55, 13, 25, 7, 17	mpl0 21983	. . . . . 6 ⊢ (𝜑 → (0g‘𝑂) = (𝐸 × {(0g‘𝑅)}))
69	67, 68	ifeq12d 4479	. . . . 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 481	. . . 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 2781	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐴‘((𝐵‘𝑋)‘𝑛)) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
72	21	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → (𝐵‘𝑋):𝐷⟶𝑆)
73		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → 𝑛 ∈ 𝐷)
74	72, 73	fvco3d 6931	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = (𝐴‘((𝐵‘𝑋)‘𝑛)))
75	37, 41	fvmpt2d 6952	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐶‘(𝐴‘𝑋))‘𝑛) = if(𝑛 = (𝐼 × {0}), (𝐴‘𝑋), (0g‘𝑂)))
76	71, 74, 75	3eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐷) → ((𝐴 ∘ (𝐵‘𝑋))‘𝑛) = ((𝐶‘(𝐴‘𝑋))‘𝑛))
77	23, 43, 76	eqfnfvd 6977	1 ⊢ (𝜑 → (𝐴 ∘ (𝐵‘𝑋)) = (𝐶‘(𝐴‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1543   ∈ wcel 2115  {crab 3388  Vcvv 3428   ⊆ wss 3886  ifcif 4457  {csn 4558   ↦ cmpt 5156   × cxp 5619  ^◡ccnv 5620   “ cima 5624   ∘ ccom 5625  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359   ↑_m cmap 8766  Fincfn 8886  0cc0 11032  ℕcn 12168  ℕ₀cn0 12431  Basecbs 17173  Scalarcsca 17217  0_gc0g 17396  Ringcrg 20208  CRingccrg 20209  AssAlgcasa 21828  algSccascl 21830   mPoly cmpl 21884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3or 1089  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3061  df-rmo 3341  df-reu 3342  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3906  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-iin 4927  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-isom 6497  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623  df-ofr 7624  df-om 7810  df-1st 7934  df-2nd 7935  df-supp 8104  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-z 12519  df-dec 12639  df-uz 12783  df-fz 13456  df-fzo 13603  df-seq 13958  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-mulr 17228  df-sca 17230  df-vsca 17231  df-ip 17232  df-tset 17233  df-ple 17234  df-ds 17236  df-hom 17238  df-cco 17239  df-0g 17398  df-gsum 17399  df-prds 17404  df-pws 17406  df-mre 17542  df-mrc 17543  df-acs 17545  df-mgm 18602  df-sgrp 18681  df-mnd 18697  df-mhm 18745  df-submnd 18746  df-grp 18906  df-minusg 18907  df-sbg 18908  df-mulg 19038  df-subg 19093  df-ghm 19182  df-cntz 19286  df-cmn 19751  df-abl 19752  df-mgp 20116  df-rng 20128  df-ur 20157  df-ring 20210  df-cring 20211  df-subrng 20521  df-subrg 20545  df-lmod 20855  df-lss 20925  df-assa 21831  df-ascl 21833  df-psr 21887  df-mpl 21889

This theorem is referenced by:  selvascl  33704