lindsrng01 - Mathbox for Alexander van der Vekens

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Theorem lindsrng01 47227

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 20489), 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 20489	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6905	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14325	. . . . . . . . . 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 20483	. . . . . . . . 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 20307	. . . . . . . 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 8835	. . . . . . . . . . . 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 7208	. . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 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 7418	. . . . . . . . . . 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 47205	. . . . . . . . 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 7411 ↑_m cmap 8822 finSupp cfsupp 9363 0cc0 11112 1c1 11113 ♯chash 14292 Basecbs 17146 Scalarcsca 17202 0_gc0g 17387 Ringcrg 20058 LModclmod 20475 linC clinc 47163 linIndS clininds 47199

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 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-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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 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-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-hash 14293 df-0g 17389 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-ring 20060 df-lmod 20477 df-lininds 47201

This theorem is referenced by: lindszr 47228

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