Theorem lindsrng01 43265

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 19268), 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 19268	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6460	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 13469	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2979	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
9	6, 8	syl5bi 234	. . . . . . . 8 ⊢ (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
11	10	adantr 474	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
12	11	com12 32	. . . . 5 ⊢ ((♯‘𝐸) = 0 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
13	1	lmodring 19263	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 474	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2777	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	2, 15	0ring 19667	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
17	14, 16	sylan 575	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
18		simpr 479	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	18	adantr 474	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantl 475	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21		snex 5140	. . . . . . . . . . . . . 14 ⊢ {(0g‘𝑅)} ∈ V
22	19, 21	jctil 515	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
23	22	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24		elmapg 8153	. . . . . . . . . . . 12 ⊢ (({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0g‘𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0g‘𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
26		fvex 6459	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) ∈ V
27	26	fconst2 6742	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0g‘𝑅)}))
28		fconstmpt 5411	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅))
29	28	eqeq2i 2789	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
30	27, 29	bitri 267	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
31		eqidd 2778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
32		eqidd 2778	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0g‘𝑅) = (0g‘𝑅))
33		simpr 479	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
34		fvexd 6461	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 6548	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
36	35	ralrimiva 3147	. . . . . . . . . . . . . 14 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
37	36	a1d 25	. . . . . . . . . . . . 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 4889	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓 finSupp (0g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅)))
39		oveq1 6929	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2779	. . . . . . . . . . . . . . 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 624	. . . . . . . . . . . . . 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 6445	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣))
43	42	eqeq1d 2779	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓‘𝑣) = (0g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
44	43	ralbidv 3167	. . . . . . . . . . . . . 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 336	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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	syl5bi 234	. . . . . . . . . . 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 232	. . . . . . . . . 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 3146	. . . . . . . . 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 6929	. . . . . . . . . . 11 ⊢ (𝐸 = {(0g‘𝑅)} → (𝐸 ↑𝑚 𝑆) = ({(0g‘𝑅)} ↑𝑚 𝑆))
51	50	raleqdv 3339	. . . . . . . . . 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 474	. . . . . . . . 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 249	. . . . . . . 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 476	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
55	54	ancomd 455	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
56	55	adantl 475	. . . . . . . . 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 2777	. . . . . . . . . 10 ⊢ (0g‘𝑀) = (0g‘𝑀)
59	57, 58, 1, 2, 15	islininds 43243	. . . . . . . . 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 17	. . . . . . . 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 703	. . . . . . 7 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 678	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
63	62	expcom 404	. . . . 5 ⊢ ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
64	12, 63	jaoi 846	. . . 4 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	64	expd 406	. . 3 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑀 ∈ LMod → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
66	65	com12 32	. 2 ⊢ (𝑀 ∈ LMod → (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
67	66	3imp 1098	1 ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 836   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ≠ wne 2968  ∀wral 3089  Vcvv 3397  ∅c0 4140  𝒫 cpw 4378  {csn 4397   class class class wbr 4886   ↦ cmpt 4965   × cxp 5353  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↑_𝑚 cmap 8140   finSupp cfsupp 8563  0cc0 10272  1c1 10273  ♯chash 13435  Basecbs 16255  Scalarcsca 16341  0_gc0g 16486  Ringcrg 18934  LModclmod 19255   linC clinc 43201   linIndS clininds 43237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-ring 18936  df-lmod 19257  df-lininds 43239

This theorem is referenced by:  lindszr  43266