Theorem cply1coe0bi 22365

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 22364	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
7	6	ad4ant13 761	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
8		fveq2 6867	. . . . . . . 8 ⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) = (coe1‘(𝐴‘𝑠)))
9	8	fveq1d 6869	. . . . . . 7 ⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛))
10	9	eqeq1d 2764	. . . . . 6 ⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
11	10	ralbidv 3185	. . . . 5 ⊢ (𝑀 = (𝐴‘𝑠) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
12	11	adantl 485	. . . 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 259	. . 3 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 )
14	13	rexlimdva2 3165	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) → ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))
15		simpr 488	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
16		0nn0 12496	. . . . . 6 ⊢ 0 ∈ ℕ0
17		eqid 2762	. . . . . . 7 ⊢ (coe1‘𝑀) = (coe1‘𝑀)
18	17, 4, 3, 1	coe1fvalcl 22274	. . . . . 6 ⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝑀)‘0) ∈ 𝐾)
19	15, 16, 18	sylancl 595	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ 𝐾)
20	19	adantr 484	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) ∈ 𝐾)
21		fveq2 6867	. . . . . 6 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0)))
22	21	eqeq2d 2773	. . . . 5 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0))))
23	22	adantl 485	. . . 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 486	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
25		eqid 2762	. . . . . . . . . 10 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
26	3	ply1ring 22309	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	3	ply1lmod 22313	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
28		eqid 2762	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	5, 25, 26, 27, 28, 4	asclf 21933	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
30	29	adantr 484	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
31		eqid 2762	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	17, 4, 3, 31	coe1fvalcl 22274	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝐵 ∧ 0 ∈ ℕ0) → ((coe1‘𝑀)‘0) ∈ (Base‘𝑅))
33	15, 16, 32	sylancl 595	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘𝑅))
34	3	ply1sca 22314	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
35	34	eqcomd 2768	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
36	35	fveq2d 6871	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
37	36	adantr 484	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
38	33, 37	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
39	30, 38	ffvelcdmd 7066	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵)
40	24, 15, 39	3jca 1141	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵))
41	40	adantr 484	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵))
42		simpr 488	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = 0 )
43	3, 5, 1, 2	coe1scl 22350	. . . . . . . . . . . . . . 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 600	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
45	44	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
46		nnne0 12247	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
47	46	neneqd 2962	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
48	47	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
49	48	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0)
50		eqeq1 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
51	50	notbid 320	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
52	51	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
53	49, 52	mpbird 259	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0)
54	53	iffalsed 4491	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 )
55		nnnn0 12488	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
56	55	adantl 485	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
57	2	fvexi 6881	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
59	45, 54, 56, 58	fvmptd 6983	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 )
60	59	eqcomd 2768	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
61	60	adantr 484	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
62	42, 61	eqtrd 2797	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
63	62	ex 416	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → (((coe1‘𝑀)‘𝑛) = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛)))
64	63	ralimdva 3174	. . . . . . 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 410	. . . . . 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 22351	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ ((coe1‘𝑀)‘0) ∈ 𝐾) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
67	19, 66	syldan 600	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
68	67	adantr 484	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
69		df-n0 12482	. . . . . . . 8 ⊢ ℕ0 = (ℕ ∪ {0})
70	69	raleqi 3318	. . . . . . 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 11173	. . . . . . . 8 ⊢ 0 ∈ V
72		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0))
73		fveq2 6867	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
74	72, 73	eqeq12d 2778	. . . . . . . . 9 ⊢ (𝑛 = 0 → (((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) ↔ ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0)))
75	74	ralunsn 4852	. . . . . . . 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 285	. . . . . 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 723	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
79		eqid 2762	. . . . . 6 ⊢ (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (coe1‘(𝐴‘((coe1‘𝑀)‘0)))
80	3, 4, 17, 79	eqcoe1ply1eq 22362	. . . . 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 3583	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠))
83	82	ex 416	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠)))
84	14, 83	impbid 214	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∪ cun 3902  ifcif 4480  {csn 4582   ↦ cmpt 5181  ⟶wf 6517  ‘cfv 6521  0cc0 11073  ℕcn 12210  ℕ₀cn0 12481  Basecbs 17245  Scalarcsca 17289  0_gc0g 17468  Ringcrg 20283  algSccascl 21904  Poly₁cpl1 22239  coe₁cco1 22240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  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 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-sup 9388  df-oi 9458  df-card 9897  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-nn 12211  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12482  df-z 12569  df-dec 12689  df-uz 12840  df-fz 13513  df-fzo 13660  df-seq 14015  df-hash 14344  df-struct 17183  df-sets 17200  df-slot 17218  df-ndx 17230  df-base 17246  df-ress 17267  df-plusg 17299  df-mulr 17300  df-sca 17302  df-vsca 17303  df-ip 17304  df-tset 17305  df-ple 17306  df-ds 17308  df-hom 17310  df-cco 17311  df-0g 17470  df-gsum 17471  df-prds 17476  df-pws 17478  df-mre 17614  df-mrc 17615  df-acs 17617  df-mgm 18674  df-sgrp 18753  df-mnd 18769  df-mhm 18817  df-submnd 18818  df-grp 18978  df-minusg 18979  df-sbg 18980  df-mulg 19110  df-subg 19165  df-ghm 19254  df-cntz 19357  df-cmn 19822  df-abl 19823  df-mgp 20187  df-rng 20199  df-ur 20232  df-srg 20237  df-ring 20285  df-subrng 20596  df-subrg 20620  df-lmod 20929  df-lss 20999  df-ascl 21907  df-psr 21961  df-mvr 21962  df-mpl 21963  df-opsr 21965  df-psr1 22242  df-vr1 22243  df-ply1 22244  df-coe1 22245

This theorem is referenced by:  cpmatel2  22773