Theorem cply1coe0bi 21831

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 21830	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑠 ∈ 𝐾) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
7	6	ad4ant13 749	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ 𝐾) ∧ 𝑀 = (𝐴‘𝑠)) → ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 )
8		fveq2 6891	. . . . . . . 8 ⊢ (𝑀 = (𝐴‘𝑠) → (coe1‘𝑀) = (coe1‘(𝐴‘𝑠)))
9	8	fveq1d 6893	. . . . . . 7 ⊢ (𝑀 = (𝐴‘𝑠) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘𝑠))‘𝑛))
10	9	eqeq1d 2734	. . . . . 6 ⊢ (𝑀 = (𝐴‘𝑠) → (((coe1‘𝑀)‘𝑛) = 0 ↔ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
11	10	ralbidv 3177	. . . . 5 ⊢ (𝑀 = (𝐴‘𝑠) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ↔ ∀𝑛 ∈ ℕ ((coe1‘(𝐴‘𝑠))‘𝑛) = 0 ))
12	11	adantl 482	. . . 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 256	. . 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 485	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
16		0nn0 12489	. . . . . 6 ⊢ 0 ∈ ℕ0
17		eqid 2732	. . . . . . 7 ⊢ (coe1‘𝑀) = (coe1‘𝑀)
18	17, 4, 3, 1	coe1fvalcl 21742	. . . . . 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 481	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) ∈ 𝐾)
21		fveq2 6891	. . . . . 6 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝐴‘𝑠) = (𝐴‘((coe1‘𝑀)‘0)))
22	21	eqeq2d 2743	. . . . 5 ⊢ (𝑠 = ((coe1‘𝑀)‘0) → (𝑀 = (𝐴‘𝑠) ↔ 𝑀 = (𝐴‘((coe1‘𝑀)‘0))))
23	22	adantl 482	. . . 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 483	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
25		eqid 2732	. . . . . . . . . 10 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
26	3	ply1ring 21777	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
27	3	ply1lmod 21781	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
28		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
29	5, 25, 26, 27, 28, 4	asclf 21442	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
30	29	adantr 481	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐴:(Base‘(Scalar‘𝑃))⟶𝐵)
31		eqid 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
32	17, 4, 3, 31	coe1fvalcl 21742	. . . . . . . . . 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 21782	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
35	34	eqcomd 2738	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (Scalar‘𝑃) = 𝑅)
36	35	fveq2d 6895	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
37	36	adantr 481	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
38	33, 37	eleqtrrd 2836	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((coe1‘𝑀)‘0) ∈ (Base‘(Scalar‘𝑃)))
39	30, 38	ffvelcdmd 7087	. . . . . . 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 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → (𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ (𝐴‘((coe1‘𝑀)‘0)) ∈ 𝐵))
42		simpr 485	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = 0 )
43	3, 5, 1, 2	coe1scl 21816	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 )))
46		nnne0 12248	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
47	46	neneqd 2945	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
48	47	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ¬ 𝑛 = 0)
49	48	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑛 = 0)
50		eqeq1 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → (𝑘 = 0 ↔ 𝑛 = 0))
51	50	notbid 317	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
52	51	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → (¬ 𝑘 = 0 ↔ ¬ 𝑛 = 0))
53	49, 52	mpbird 256	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → ¬ 𝑘 = 0)
54	53	iffalsed 4539	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ 𝑘 = 𝑛) → if(𝑘 = 0, ((coe1‘𝑀)‘0), 0 ) = 0 )
55		nnnn0 12481	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
56	55	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0)
57	2	fvexi 6905	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
58	57	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 ∈ V)
59	45, 54, 56, 58	fvmptd 7005	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = 0 )
60	59	eqcomd 2738	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
61	60	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → 0 = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
62	42, 61	eqtrd 2772	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ) ∧ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
63	62	ex 413	. . . . . . . 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 407	. . . . . 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 21817	. . . . . . . 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 481	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ((coe1‘𝑀)‘0) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
69		df-n0 12475	. . . . . . . 8 ⊢ ℕ0 = (ℕ ∪ {0})
70	69	raleqi 3323	. . . . . . 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 11210	. . . . . . . 8 ⊢ 0 ∈ V
72		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘𝑀)‘𝑛) = ((coe1‘𝑀)‘0))
73		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑛 = 0 → ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘0))
74	72, 73	eqeq12d 2748	. . . . . . . . 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 282	. . . . . 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 711	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∀𝑛 ∈ ℕ0 ((coe1‘𝑀)‘𝑛) = ((coe1‘(𝐴‘((coe1‘𝑀)‘0)))‘𝑛))
79		eqid 2732	. . . . . 6 ⊢ (coe1‘(𝐴‘((coe1‘𝑀)‘0))) = (coe1‘(𝐴‘((coe1‘𝑀)‘0)))
80	3, 4, 17, 79	eqcoe1ply1eq 21828	. . . . 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 3614	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ) → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠))
83	82	ex 413	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 → ∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠)))
84	14, 83	impbid 211	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (∃𝑠 ∈ 𝐾 𝑀 = (𝐴‘𝑠) ↔ ∀𝑛 ∈ ℕ ((coe1‘𝑀)‘𝑛) = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∪ cun 3946  ifcif 4528  {csn 4628   ↦ cmpt 5231  ⟶wf 6539  ‘cfv 6543  0cc0 11112  ℕcn 12214  ℕ₀cn0 12474  Basecbs 17146  Scalarcsca 17202  0_gc0g 17387  Ringcrg 20058  algSccascl 21413  Poly₁cpl1 21707  coe₁cco1 21708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-ofr 7673  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12475  df-z 12561  df-dec 12680  df-uz 12825  df-fz 13487  df-fzo 13630  df-seq 13969  df-hash 14293  df-struct 17082  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-ip 17217  df-tset 17218  df-ple 17219  df-ds 17221  df-hom 17223  df-cco 17224  df-0g 17389  df-gsum 17390  df-prds 17395  df-pws 17397  df-mre 17532  df-mrc 17533  df-acs 17535  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-mulg 18953  df-subg 19005  df-ghm 19092  df-cntz 19183  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-srg 20012  df-ring 20060  df-subrg 20321  df-lmod 20477  df-lss 20548  df-ascl 21416  df-psr 21468  df-mvr 21469  df-mpl 21470  df-opsr 21472  df-psr1 21710  df-vr1 21711  df-ply1 21712  df-coe1 21713

This theorem is referenced by:  cpmatel2  22222