Theorem cply1coe0bi 22306

Description: A polynomial is constant (i.e. a "lifted scalar") iff all but the first coefficient are zero. (Contributed by AV, 16-Nov-2019.)

Hypotheses

Ref	Expression
cply1coe0.k	⊢ 𝐾 = (Base‘𝑅)
cply1coe0.0	⊢ 0 = (0g‘𝑅)
cply1coe0.p	⊢ 𝑃 = (Poly1‘𝑅)
cply1coe0.b	⊢ 𝐵 = (Base‘𝑃)
cply1coe0.a	⊢ 𝐴 = (algSc‘𝑃)

Assertion

Ref	Expression
cply1coe0bi	⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))

Distinct variable groups:   𝑛,𝐾   𝑅,𝑛   𝐴,𝑛,𝑠   𝐵,𝑛,𝑠   𝐾,𝑠   𝑛,𝑀,𝑠   𝑅,𝑠   0 ,𝑠

Allowed substitution hints:   𝑃(𝑛,𝑠)   0 (𝑛)

Proof of Theorem cply1coe0bi

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cply1coe0.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝑅)
2		cply1coe0.0	. . . . . 6 ⊢ 0 = (0g‘𝑅)
3		cply1coe0.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
4		cply1coe0.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
5		cply1coe0.a	. . . . . 6 ⊢ 𝐴 = (algSc‘𝑃)
6	1, 2, 3, 4, 5	cply1coe0 22305	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
7	6	ad4ant13 751	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
8		fveq2 6906	. . . . . . . 8 ⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) = (coe1‘(𝐴‘𝑠)))
9	8	fveq1d 6908	. . . . . . 7 ⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛))
10	9	eqeq1d 2739	. . . . . 6 ⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
11	10	ralbidv 3178	. . . . 5 ⊢ (𝑀 = (𝐴‘𝑠) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
12	11	adantl 481	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
13	7, 12	mpbird 257	. . 3 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )
14	13	rexlimdva2 3157	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))
15		simpr 484	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
16		0nn0 12541	. . . . . 6 ⊢ 0 ∈ ℕ0
17		eqid 2737	. . . . . . 7 ⊢ (coe1‘𝑀) = (coe1‘𝑀)
18	17, 4, 3, 1	coe1fvalcl 22214	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝑀)‘0) ∈ 𝐾)
19	15, 16, 18	sylancl 586	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ 𝐾)
20	19	adantr 480	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) ∈ 𝐾)
21		fveq2 6906	. . . . . 6 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0)))
22	21	eqeq2d 2748	. . . . 5 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0))))
23	22	adantl 481	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) ∧ 𝑠 = ((coe1‘𝑀)‘0)) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0))))
24		simpl 482	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
25		eqid 2737	. . . . . . . . . 10 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
26	3	ply1ring 22249	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	3	ply1lmod 22253	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
28		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	5, 25, 26, 27, 28, 4	asclf 21902	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
31		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	17, 4, 3, 31	coe1fvalcl 22214	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝑀)‘0) ∈ (Base‘𝑅))
33	15, 16, 32	sylancl 586	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘𝑅))
34	3	ply1sca 22254	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
35	34	eqcomd 2743	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
36	35	fveq2d 6910	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
37	36	adantr 480	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
38	33, 37	eleqtrrd 2844	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
39	30, 38	ffvelcdmd 7105	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)
40	24, 15, 39	3jca 1129	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵))
41	40	adantr 480	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵))
42		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = 0 )
43	3, 5, 1, 2	coe1scl 22290	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝑀)‘0) ∈ 𝐾) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
44	19, 43	syldan 591	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
46		nnne0 12300	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
47	46	neneqd 2945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
48	47	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
49	48	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0)
50		eqeq1 2741	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
51	50	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
52	51	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
53	49, 52	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0)
54	53	iffalsed 4536	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 )
55		nnnn0 12533	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
56	55	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
57	2	fvexi 6920	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
59	45, 54, 56, 58	fvmptd 7023	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 )
60	59	eqcomd 2743	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
62	42, 61	eqtrd 2777	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
63	62	ex 412	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → (((coe1‘𝑀)‘𝑛) = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)))
64	63	ralimdva 3167	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)))
65	64	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
66	3, 5, 1	ply1sclid 22291	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝑀)‘0) ∈ 𝐾) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
67	19, 66	syldan 591	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
68	67	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
69		df-n0 12527	. . . . . . . 8 ⊢ ℕ0 = (ℕ ∪ {0})
70	69	raleqi 3324	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ∀𝑛 ∈ (ℕ ∪ {0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
71		c0ex 11255	. . . . . . . 8 ⊢ 0 ∈ V
72		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0))
73		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
74	72, 73	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑛 = 0 → (((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))
75	74	ralunsn 4894	. . . . . . . 8 ⊢ (0 ∈ V → (∀𝑛 ∈ (ℕ ∪ {0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))))
76	71, 75	mp1i 13	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ (ℕ ∪ {0})((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))))
77	70, 76	bitrid 283	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ∧ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))))
78	65, 68, 77	mpbir2and 713	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
79		eqid 2737	. . . . . 6 ⊢ (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (coe1‘(𝐴‘((coe1‘𝑀)‘0)))
80	3, 4, 17, 79	eqcoe1ply1eq 22303	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵) → (∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) → 𝑀 = (𝐴‘((coe1‘𝑀)‘0))))
81	41, 78, 80	sylc 65	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → 𝑀 = (𝐴‘((coe1‘𝑀)‘0)))
82	20, 23, 81	rspcedvd 3624	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠))
83	82	ex 412	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠)))
84	14, 83	impbid 212	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∪ cun 3949  ifcif 4525  {csn 4626   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  0cc0 11155  ℕcn 12266  ℕ₀cn0 12526  Basecbs 17247  Scalarcsca 17300  0_gc0g 17484  Ringcrg 20230  algSccascl 21872  Poly₁cpl1 22178  coe₁cco1 22179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184

This theorem is referenced by:  cpmatel2  22719