lindsrng01 - Mathbox for Alexander van der Vekens

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Theorem lindsrng01 47139

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 20484), 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 20484	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6905	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14322	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2951	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
9	6, 8	biimtrid 241	. . . . . . . 8 ⊢ (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
11	10	adantr 481	. . . . . 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 20478	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 481	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2732	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
16	2, 15	0ring 20302	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0_g‘𝑅)})
17	14, 16	sylan 580	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0_g‘𝑅)})
18		simpr 485	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	18	adantr 481	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantl 482	. . . . . . . 8 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21		snex 5431	. . . . . . . . . . . . . 14 ⊢ {(0_g‘𝑅)} ∈ V
22	19, 21	jctil 520	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0_g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
23	22	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0_g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24		elmapg 8832	. . . . . . . . . . . 12 ⊢ (({(0_g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆) ↔ 𝑓:𝑆⟶{(0_g‘𝑅)}))
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆) ↔ 𝑓:𝑆⟶{(0_g‘𝑅)}))
26		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) ∈ V
27	26	fconst2 7205	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0_g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0_g‘𝑅)}))
28		fconstmpt 5738	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0_g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))
29	28	eqeq2i 2745	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0_g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)))
30	27, 29	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0_g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)))
31		eqidd 2733	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)))
32		eqidd 2733	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0_g‘𝑅) = (0_g‘𝑅))
33		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
34		fvexd 6906	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0_g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 7005	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅))
36	35	ralrimiva 3146	. . . . . . . . . . . . . 14 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅))
37	36	a1d 25	. . . . . . . . . . . . 13 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) finSupp (0_g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅)))
38		breq1 5151	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓 finSupp (0_g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) finSupp (0_g‘𝑅)))
39		oveq1 7415	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀) ↔ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆) = (0_g‘𝑀)))
41	38, 40	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → ((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) ↔ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) finSupp (0_g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆) = (0_g‘𝑀))))
42		fveq1 6890	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣))
43	42	eqeq1d 2734	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → ((𝑓‘𝑣) = (0_g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅)))
44	43	ralbidv 3177	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅) ↔ ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅)))
45	41, 44	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)) ↔ (((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) finSupp (0_g‘𝑅) ∧ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅))))
46	37, 45	syl5ibrcom 246	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → ((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅))))
47	30, 46	biimtrid 241	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓:𝑆⟶{(0_g‘𝑅)} → ((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅))))
48	25, 47	sylbid 239	. . . . . . . . . 10 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆) → ((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅))))
49	48	ralrimiv 3145	. . . . . . . . 9 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)))
50		oveq1 7415	. . . . . . . . . . 11 ⊢ (𝐸 = {(0_g‘𝑅)} → (𝐸 ↑_m 𝑆) = ({(0_g‘𝑅)} ↑_m 𝑆))
51	50	raleqdv 3325	. . . . . . . . . 10 ⊢ (𝐸 = {(0_g‘𝑅)} → (∀𝑓 ∈ (𝐸 ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)) ↔ ∀𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅))))
52	51	adantr 481	. . . . . . . . 9 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (∀𝑓 ∈ (𝐸 ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)) ↔ ∀𝑓 ∈ ({(0_g‘𝑅)} ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅))))
53	49, 52	mpbird 256	. . . . . . . 8 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ∀𝑓 ∈ (𝐸 ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)))
54		simpl 483	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
55	54	ancomd 462	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
56	55	adantl 482	. . . . . . . . 9 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
57		lindsrng01.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
58		eqid 2732	. . . . . . . . . 10 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
59	57, 58, 1, 2, 15	islininds 47117	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)))))
60	56, 59	syl 17	. . . . . . . 8 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸 ↑_m 𝑆)((𝑓 finSupp (0_g‘𝑅) ∧ (𝑓( linC ‘𝑀)𝑆) = (0_g‘𝑀)) → ∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0_g‘𝑅)))))
61	20, 53, 60	mpbir2and 711	. . . . . . 7 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 686	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
63	62	expcom 414	. . . . 5 ⊢ ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
64	12, 63	jaoi 855	. . . 4 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	64	expd 416	. . 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 1111	1 ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 Vcvv 3474 ∅c0 4322 𝒫 cpw 4602 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819 finSupp cfsupp 9360 0cc0 11109 1c1 11110 ♯chash 14289 Basecbs 17143 Scalarcsca 17199 0_gc0g 17384 Ringcrg 20055 LModclmod 20470 linC clinc 47075 linIndS clininds 47111

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-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

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-op 4635 df-uni 4909 df-int 4951 df-iun 4999 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-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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-card 9933 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-uz 12822 df-fz 13484 df-hash 14290 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-ring 20057 df-lmod 20472 df-lininds 47113

This theorem is referenced by: lindszr 47140

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