Theorem lindsrng01 42051

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 18699), 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 18699	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4		fvex 6113	. . . . . . . . . . 11 ⊢ (Base‘𝑅) ∈ V
5	2, 4	eqeltri 2684	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
6		hasheq0 13015	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((#‘𝐸) = 0 ↔ 𝐸 = ∅))
7	5, 6	ax-mp 5	. . . . . . . . 9 ⊢ ((#‘𝐸) = 0 ↔ 𝐸 = ∅)
8		eqneqall 2793	. . . . . . . . . 10 ⊢ (𝐸 = ∅ → (𝐸 ≠ ∅ → 𝑆 linIndS 𝑀))
9	8	com12 32	. . . . . . . . 9 ⊢ (𝐸 ≠ ∅ → (𝐸 = ∅ → 𝑆 linIndS 𝑀))
10	7, 9	syl5bi 231	. . . . . . . 8 ⊢ (𝐸 ≠ ∅ → ((#‘𝐸) = 0 → 𝑆 linIndS 𝑀))
11	3, 10	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ LMod → ((#‘𝐸) = 0 → 𝑆 linIndS 𝑀))
12	11	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → ((#‘𝐸) = 0 → 𝑆 linIndS 𝑀))
13	12	com12 32	. . . . 5 ⊢ ((#‘𝐸) = 0 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
14	1	lmodring 18694	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
16		eqid 2610	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
17	2, 16	0ring 19091	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (#‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
18	15, 17	sylan 487	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → 𝐸 = {(0g‘𝑅)})
19		simpr 476	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantr 480	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
21	20	adantl 481	. . . . . . . 8 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
22		snex 4835	. . . . . . . . . . . . . 14 ⊢ {(0g‘𝑅)} ∈ V
23	20, 22	jctil 558	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24	23	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → ({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
25		elmapg 7757	. . . . . . . . . . . 12 ⊢ (({(0g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑓 ∈ ({(0g‘𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
26	24, 25	syl 17	. . . . . . . . . . 11 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → (𝑓 ∈ ({(0g‘𝑅)} ↑𝑚 𝑆) ↔ 𝑓:𝑆⟶{(0g‘𝑅)}))
27		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) ∈ V
28	27	fconst2 6375	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0g‘𝑅)}))
29		fconstmpt 5085	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅))
30	29	eqeq2i 2622	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
31	28, 30	bitri 263	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
32		eqidd 2611	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)))
33		eqidd 2611	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0g‘𝑅) = (0g‘𝑅))
34		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
35	27	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0g‘𝑅) ∈ V)
36	32, 33, 34, 35	fvmptd 6197	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
37	36	ralrimiva 2949	. . . . . . . . . . . . . 14 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅))
38	37	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)))
39		breq1 4586	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓 finSupp (0g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) finSupp (0g‘𝑅)))
40		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆))
41	40	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓( linC ‘𝑀)𝑆) = (0g‘𝑀) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))( linC ‘𝑀)𝑆) = (0g‘𝑀)))
42	39, 41	anbi12d 743	. . . . . . . . . . . . . 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))))
43		fveq1 6102	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣))
44	43	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → ((𝑓‘𝑣) = (0g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
45	44	ralbidv 2969	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0g‘𝑅)) → (∀𝑣 ∈ 𝑆 (𝑓‘𝑣) = (0g‘𝑅) ↔ ∀𝑣 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (0g‘𝑅))‘𝑣) = (0g‘𝑅)))
46	42, 45	imbi12d 333	. . . . . . . . . . . . 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))))
47	38, 46	syl5ibrcom 236	. . . . . . . . . . . 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))))
48	31, 47	syl5bi 231	. . . . . . . . . . 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))))
49	26, 48	sylbid 229	. . . . . . . . . 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))))
50	49	ralrimiv 2948	. . . . . . . . 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)))
51		oveq1 6556	. . . . . . . . . . 11 ⊢ (𝐸 = {(0g‘𝑅)} → (𝐸 ↑𝑚 𝑆) = ({(0g‘𝑅)} ↑𝑚 𝑆))
52	51	raleqdv 3121	. . . . . . . . . 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))))
53	52	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))))
54	50, 53	mpbird 246	. . . . . . . 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)))
55		simpl 472	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
56	55	ancomd 466	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
57	56	adantl 481	. . . . . . . . 9 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
58		lindsrng01.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑀)
59		eqid 2610	. . . . . . . . . 10 ⊢ (0g‘𝑀) = (0g‘𝑀)
60	58, 59, 1, 2, 16	islininds 42029	. . . . . . . . 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)))))
61	57, 60	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)))))
62	21, 54, 61	mpbir2and 959	. . . . . . 7 ⊢ ((𝐸 = {(0g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
63	18, 62	mpancom 700	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (#‘𝐸) = 1) → 𝑆 linIndS 𝑀)
64	63	expcom 450	. . . . 5 ⊢ ((#‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	13, 64	jaoi 393	. . . 4 ⊢ (((#‘𝐸) = 0 ∨ (#‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
66	65	expd 451	. . 3 ⊢ (((#‘𝐸) = 0 ∨ (#‘𝐸) = 1) → (𝑀 ∈ LMod → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
67	66	com12 32	. 2 ⊢ (𝑀 ∈ LMod → (((#‘𝐸) = 0 ∨ (#‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 → 𝑆 linIndS 𝑀)))
68	67	3imp 1249	1 ⊢ ((𝑀 ∈ LMod ∧ ((#‘𝐸) = 0 ∨ (#‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744   finSupp cfsupp 8158  0cc0 9815  1c1 9816  #chash 12979  Basecbs 15695  Scalarcsca 15771  0_gc0g 15923  Ringcrg 18370  LModclmod 18686   linC clinc 41987   linIndS clininds 42023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-ring 18372  df-lmod 18688  df-lininds 42025

This theorem is referenced by:  lindszr  42052