Theorem lindsrng01 45697

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 20051), 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 20051	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6770	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14006	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2953	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
8	7	com12 32	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
9	6, 8	syl5bi 241	. . . . . . . 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 20046	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2738	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
16	2, 15	0ring 20454	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
17	14, 16	sylan 579	. . . . . . 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 5349	. . . . . . . . . . . . . 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 8586	. . . . . . . . . . . 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 6769	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) ∈ V
27	26	fconst2 7062	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0g‘𝑅)}))
28		fconstmpt 5640	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅))
29	28	eqeq2i 2751	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
30	27, 29	bitri 274	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
31		eqidd 2739	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
32		eqidd 2739	. . . . . . . . . . . . . . . 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 6771	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 6864	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
36	35	ralrimiva 3107	. . . . . . . . . . . . . 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 5073	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓 finSupp (0g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅)))
39		oveq1 7262	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2740	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . 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 6755	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣))
43	42	eqeq1d 2740	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓‘𝑣) = (0g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
44	43	ralbidv 3120	. . . . . . . . . . . . . 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 246	. . . . . . . . . . . 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 241	. . . . . . . . . . 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 239	. . . . . . . . . 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 3106	. . . . . . . . 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 7262	. . . . . . . . . . 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 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 256	. . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (0g‘𝑀) = (0g‘𝑀)
59	57, 58, 1, 2, 15	islininds 45675	. . . . . . . . 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 709	. . . . . . 7 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 684	. . . . . 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 853	. . . 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 1109	1 ⊢ ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  Vcvv 3422  ∅c0 4253  𝒫 cpw 4530  {csn 4558   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573   finSupp cfsupp 9058  0cc0 10802  1c1 10803  ♯chash 13972  Basecbs 16840  Scalarcsca 16891  0_gc0g 17067  Ringcrg 19698  LModclmod 20038   linC clinc 45633   linIndS clininds 45669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-hash 13973  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-ring 19700  df-lmod 20040  df-lininds 45671

This theorem is referenced by:  lindszr  45698