Theorem cply1coe0bi 22196

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 22195	. . . . 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 6861	. . . . . . . 8 ⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) = (coe1‘(𝐴‘𝑠)))
9	8	fveq1d 6863	. . . . . . 7 ⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛))
10	9	eqeq1d 2732	. . . . . 6 ⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
11	10	ralbidv 3157	. . . . 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 3137	. 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 12464	. . . . . 6 ⊢ 0 ∈ ℕ0
17		eqid 2730	. . . . . . 7 ⊢ (coe1‘𝑀) = (coe1‘𝑀)
18	17, 4, 3, 1	coe1fvalcl 22104	. . . . . 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 6861	. . . . . 6 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0)))
22	21	eqeq2d 2741	. . . . 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 2730	. . . . . . . . . 10 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
26	3	ply1ring 22139	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	3	ply1lmod 22143	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
28		eqid 2730	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	5, 25, 26, 27, 28, 4	asclf 21798	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
30	29	adantr 480	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
31		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	17, 4, 3, 31	coe1fvalcl 22104	. . . . . . . . . 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 22144	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
35	34	eqcomd 2736	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
36	35	fveq2d 6865	. . . . . . . . . 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 2832	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
39	30, 38	ffvelcdmd 7060	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)
40	24, 15, 39	3jca 1128	. . . . . 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 22180	. . . . . . . . . . . . . . 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 12227	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
47	46	neneqd 2931	. . . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . 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 4502	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 )
55		nnnn0 12456	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
56	55	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
57	2	fvexi 6875	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
59	45, 54, 56, 58	fvmptd 6978	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 )
60	59	eqcomd 2736	. . . . . . . . . . 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 2765	. . . . . . . . 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 3146	. . . . . . 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 22181	. . . . . . . 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 12450	. . . . . . . 8 ⊢ ℕ0 = (ℕ ∪ {0})
70	69	raleqi 3299	. . . . . . 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 11175	. . . . . . . 8 ⊢ 0 ∈ V
72		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0))
73		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
74	72, 73	eqeq12d 2746	. . . . . . . . 9 ⊢ (𝑛 = 0 → (((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))
75	74	ralunsn 4861	. . . . . . . 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 2730	. . . . . 6 ⊢ (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (coe1‘(𝐴‘((coe1‘𝑀)‘0)))
80	3, 4, 17, 79	eqcoe1ply1eq 22193	. . . . 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 3593	. . 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∪ cun 3915  ifcif 4491  {csn 4592   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  0cc0 11075  ℕcn 12193  ℕ₀cn0 12449  Basecbs 17186  Scalarcsca 17230  0_gc0g 17409  Ringcrg 20149  algSccascl 21768  Poly₁cpl1 22068  coe₁cco1 22069

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-srg 20103  df-ring 20151  df-subrng 20462  df-subrg 20486  df-lmod 20775  df-lss 20845  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074

This theorem is referenced by:  cpmatel2  22607