Theorem lindsrng01 49549

Description: Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 21142), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)

Hypotheses

Ref	Expression
lindsrng01.b	⊢ 𝐵 = (Base‘𝑀)
lindsrng01.r	⊢ 𝑅 = (Scalar‘𝑀)
lindsrng01.e	⊢ 𝐸 = (Base‘𝑅)

Assertion

Ref	Expression
lindsrng01	⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Proof of Theorem lindsrng01

Dummy variables 𝑓 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lindsrng01.r	. . . . . . . . 9 ⊢ 𝑅 = (Scalar‘𝑀)
2		lindsrng01.e	. . . . . . . . 9 ⊢ 𝐸 = (Base‘𝑅)
3	1, 2	lmodsn0 21142	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6897	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14500	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2967	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
8	7	com12 33	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
9	6, 8	biimtrid 245	. . . . . . . 8 ⊢ (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
10	3, 9	syl 18	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
11	10	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
12	11	com12 33	. . . . 5 ⊢ ((♯‘𝐸) = 0 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
13	1	lmodring 21136	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 486	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2761	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	2, 15	0ring 20770	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
17	14, 16	sylan 592	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
18		simpr 490	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	18	adantr 486	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21		snex 5397	. . . . . . . . . . . . . 14 ⊢ {(0g‘𝑅)} ∈ V
22	19, 21	jctil 529	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
23	22	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24		elmapg 8852	. . . . . . . . . . . 12 ⊢ (({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
25	23, 24	syl 18	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
26		fvex 6896	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) ∈ V
27	26	fconst2 7209	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0g‘𝑅)}))
28		fconstmpt 5713	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅))
29	28	eqeq2i 2774	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
30	27, 29	bitri 278	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
31		eqidd 2762	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
32		eqidd 2762	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0g‘𝑅) = (0g‘𝑅))
33		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
34		fvexd 6898	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 6999	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
36	35	ralrimiva 3155	. . . . . . . . . . . . . 14 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
37	36	a1d 26	. . . . . . . . . . . . 13 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (((𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
38		breq1 5106	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓 finSupp (0g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅)))
39		oveq1 7425	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆) = (0g‘𝑀)))
41	38, 40	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆) = (0g‘𝑀))))
42		fveq1 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣))
43	42	eqeq1d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓‘𝑣) = (0g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
44	43	ralbidv 3186	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅) ↔ ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
45	41, 44	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)) ↔ (((𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))))
46	37, 45	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅))))
47	30, 46	biimtrid 245	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓:𝑆⟶{(0g‘𝑅)} → ((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅))))
48	25, 47	sylbid 243	. . . . . . . . . 10 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆) → ((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅))))
49	48	ralrimiv 3154	. . . . . . . . 9 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)))
50		oveq1 7425	. . . . . . . . . . 11 ⊢ (𝐸 = {(0g‘𝑅)} → (𝐸 ↑m 𝑆) = ({(0g‘𝑅)} ↑m 𝑆))
51	50	raleqdv 3320	. . . . . . . . . 10 ⊢ (𝐸 = {(0g‘𝑅)} → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)) ↔ ∀𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅))))
52	51	adantr 486	. . . . . . . . 9 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)) ↔ ∀𝑓 ∈ ({(0g‘𝑅)} ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅))))
53	49, 52	mpbird 260	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)))
54		simpl 488	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
55	54	ancomd 467	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
56	55	adantl 487	. . . . . . . . 9 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
57		lindsrng01.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
58		eqid 2761	. . . . . . . . . 10 ⊢ (0g‘𝑀) = (0g‘𝑀)
59	57, 58, 1, 2, 15	islininds 49527	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)))))
60	56, 59	syl 18	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp (0g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅)))))
61	20, 53, 60	mpbir2and 726	. . . . . . 7 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 701	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
63	62	expcom 419	. . . . 5 ⊢ ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
64	12, 63	jaoi 871	. . . 4 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	64	expd 421	. . 3 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑀 ∈ LMod → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
66	65	com12 33	. 2 ⊢ (𝑀 ∈ LMod → (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
67	66	3imp 1128	1 ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  Vcvv 3451  ∅c0 4279  𝒫 cpw 4557  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ↑_m cmap 8840   finSupp cfsupp 9346  0cc0 11193  1c1 11194  ♯chash 14467  Basecbs 17380  Scalarcsca 17424  0_gc0g 17603  Ringcrg 20452  LModclmod 21128   linC clinc 49485   linIndS clininds 49521

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-fz 13633  df-hash 14468  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-ring 20454  df-lmod 21130  df-lininds 49523

This theorem is used by:  lindszr  49550