lindsrng01 - Mathbox for Alexander van der Vekens

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Theorem lindsrng01 47149

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 20485), 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 20485	. . . . . . . 8 ⊢ (𝑀 ∈ LMod → 𝐸 ≠ ∅)
4	2	fvexi 6906	. . . . . . . . . 10 ⊢ 𝐸 ∈ V
5		hasheq0 14323	. . . . . . . . . 10 ⊢ (𝐸 ∈ V → ((♯‘𝐸) = 0 ↔ 𝐸 = ∅))
6	4, 5	ax-mp 5	. . . . . . . . 9 ⊢ ((♯‘𝐸) = 0 ↔ 𝐸 = ∅)
7		eqneqall 2952	. . . . . . . . . 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 482	. . . . . 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 20479	. . . . . . . . 9 ⊢ (𝑀 ∈ LMod → 𝑅 ∈ Ring)
14	13	adantr 482	. . . . . . . 8 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑅 ∈ Ring)
15		eqid 2733	. . . . . . . . 9 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
16	2, 15	0ring 20303	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐸) = 1) → 𝐸 = {(0_g‘𝑅)})
17	14, 16	sylan 581	. . . . . . 7 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝐸 = {(0_g‘𝑅)})
18		simpr 486	. . . . . . . . . 10 ⊢ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	18	adantr 482	. . . . . . . . 9 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 ∈ 𝒫 𝐵)
20	19	adantl 483	. . . . . . . 8 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 ∈ 𝒫 𝐵)
21		snex 5432	. . . . . . . . . . . . . 14 ⊢ {(0_g‘𝑅)} ∈ V
22	19, 21	jctil 521	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → ({(0_g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
23	22	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → ({(0_g‘𝑅)} ∈ V ∧ 𝑆 ∈ 𝒫 𝐵))
24		elmapg 8833	. . . . . . . . . . . 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 6905	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝑅) ∈ V
27	26	fconst2 7206	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑆⟶{(0_g‘𝑅)} ↔ 𝑓 = (𝑆 × {(0_g‘𝑅)}))
28		fconstmpt 5739	. . . . . . . . . . . . . 14 ⊢ (𝑆 × {(0_g‘𝑅)}) = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))
29	28	eqeq2i 2746	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑆 × {(0_g‘𝑅)}) ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)))
30	27, 29	bitri 275	. . . . . . . . . . . 12 ⊢ (𝑓:𝑆⟶{(0_g‘𝑅)} ↔ 𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)))
31		eqidd 2734	. . . . . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) ∧ 𝑥 = 𝑣) → (0_g‘𝑅) = (0_g‘𝑅))
33		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
34		fvexd 6907	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) ∧ 𝑣 ∈ 𝑆) → (0_g‘𝑅) ∈ V)
35	31, 32, 33, 34	fvmptd 7006	. . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . 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 5152	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓 finSupp (0_g‘𝑅) ↔ (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) finSupp (0_g‘𝑅)))
39		oveq1 7416	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓( linC ‘𝑀)𝑆) = ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))( linC ‘𝑀)𝑆))
40	39	eqeq1d 2735	. . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → (𝑓‘𝑣) = ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣))
43	42	eqeq1d 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑥 ∈ 𝑆 ↦ (0_g‘𝑅)) → ((𝑓‘𝑣) = (0_g‘𝑅) ↔ ((𝑥 ∈ 𝑆 ↦ (0_g‘𝑅))‘𝑣) = (0_g‘𝑅)))
44	43	ralbidv 3178	. . . . . . . . . . . . . 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 345	. . . . . . . . . . . . 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 3146	. . . . . . . . 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 7416	. . . . . . . . . . 11 ⊢ (𝐸 = {(0_g‘𝑅)} → (𝐸 ↑_m 𝑆) = ({(0_g‘𝑅)} ↑_m 𝑆))
51	50	raleqdv 3326	. . . . . . . . . 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 482	. . . . . . . . 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 257	. . . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵))
55	54	ancomd 463	. . . . . . . . . 10 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → (𝑆 ∈ 𝒫 𝐵 ∧ 𝑀 ∈ LMod))
56	55	adantl 483	. . . . . . . . 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 2733	. . . . . . . . . 10 ⊢ (0_g‘𝑀) = (0_g‘𝑀)
59	57, 58, 1, 2, 15	islininds 47127	. . . . . . . . 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 712	. . . . . . 7 ⊢ ((𝐸 = {(0_g‘𝑅)} ∧ ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1)) → 𝑆 linIndS 𝑀)
62	17, 61	mpancom 687	. . . . . 6 ⊢ (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (♯‘𝐸) = 1) → 𝑆 linIndS 𝑀)
63	62	expcom 415	. . . . 5 ⊢ ((♯‘𝐸) = 1 → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
64	12, 63	jaoi 856	. . . 4 ⊢ (((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) → ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀))
65	64	expd 417	. . 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 1112	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 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∅c0 4323 𝒫 cpw 4603 {csn 4629 class class class wbr 5149 ↦ cmpt 5232 × cxp 5675 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_m cmap 8820 finSupp cfsupp 9361 0cc0 11110 1c1 11111 ♯chash 14290 Basecbs 17144 Scalarcsca 17200 0_gc0g 17385 Ringcrg 20056 LModclmod 20471 linC clinc 47085 linIndS clininds 47121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-grp 18822 df-ring 20058 df-lmod 20473 df-lininds 47123

This theorem is referenced by: lindszr 47150

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