Theorem lindsrng01 48714

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 20825), 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 20825	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6848	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14286	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2943	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
9	6, 8	biimtrid 242	. . . . . . . 8 ⊢ (𝐸 ≠ ∅ → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ((♯‘𝐸) = 0 → 𝑆 linIndS 𝑀))
11	10	adantr 480	. . . . . 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 20819	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2736	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	2, 15	0ring 20459	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
17	14, 16	sylan 580	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	18	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21		snex 5381	. . . . . . . . . . . . . 14 ⊢ {(0g‘𝑅)} ∈ V
22	19, 21	jctil 519	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
23	22	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24		elmapg 8776	. . . . . . . . . . . 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 6847	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) ∈ V
27	26	fconst2 7151	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0g‘𝑅)}))
28		fconstmpt 5686	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅))
29	28	eqeq2i 2749	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
30	27, 29	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
31		eqidd 2737	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
32		eqidd 2737	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0g‘𝑅) = (0g‘𝑅))
33		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
34		fvexd 6849	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 6948	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
36	35	ralrimiva 3128	. . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓 finSupp (0g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅)))
39		oveq1 7365	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2738	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 6833	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣))
43	42	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓‘𝑣) = (0g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
44	43	ralbidv 3159	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 240	. . . . . . . . . 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 3127	. . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝐸 = {(0g‘𝑅)} → (𝐸 ↑m 𝑆) = ({(0g‘𝑅)} ↑m 𝑆))
51	50	raleqdv 3296	. . . . . . . . . 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 480	. . . . . . . . 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 257	. . . . . . . 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 482	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
55	54	ancomd 461	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
56	55	adantl 481	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (0g‘𝑀) = (0g‘𝑀)
59	57, 58, 1, 2, 15	islininds 48692	. . . . . . . . 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 713	. . . . . . 7 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 688	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
63	62	expcom 413	. . . . 5 ⊢ ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
64	12, 63	jaoi 857	. . . 4 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	64	expd 415	. . 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 1110	1 ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  Vcvv 3440  ∅c0 4285  𝒫 cpw 4554  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763   finSupp cfsupp 9264  0cc0 11026  1c1 11027  ♯chash 14253  Basecbs 17136  Scalarcsca 17180  0_gc0g 17359  Ringcrg 20168  LModclmod 20811   linC clinc 48650   linIndS clininds 48686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9851  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-hash 14254  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-ring 20170  df-lmod 20813  df-lininds 48688

This theorem is referenced by:  lindszr  48715