ldepsnlinc - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinc 48996

Description: The reverse implication of islindeps2 48971 does not hold for arbitrary (left) modules, see note in [Roman] p. 112: "... if a nontrivial linear combination of the elements ... in an R-module M is 0, ... where not all of the coefficients are 0, then we cannot conclude ... that one of the elements ... is a linear combination of the others." This means that there is at least one left module having a linearly dependent subset in which there is at least one element which is not a linear combination of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 48984 and zlmodzxznm 48985. (Contributed by AV, 25-May-2019.) (Revised by AV, 30-Jul-2019.)

**Assertion**
Ref	Expression
ldepsnlinc	⊢ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣))

Distinct variable group: 𝑓,𝑚,𝑠,𝑣

**Proof of Theorem ldepsnlinc**
Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (ℤ_ring freeLMod {0, 1}) = (ℤ_ring freeLMod {0, 1})
2	1	zlmodzxzlmod 48842	. . 3 ⊢ ((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1})))
3	2	simpli 483	. 2 ⊢ (ℤ_ring freeLMod {0, 1}) ∈ LMod
4		3z 12551	. . . . 5 ⊢ 3 ∈ ℤ
5		6nn 12261	. . . . . 6 ⊢ 6 ∈ ℕ
6	5	nnzi 12542	. . . . 5 ⊢ 6 ∈ ℤ
7	1	zlmodzxzel 48843	. . . . 5 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
8	4, 6, 7	mp2an 693	. . . 4 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
9		2z 12550	. . . . 5 ⊢ 2 ∈ ℤ
10		4z 12552	. . . . 5 ⊢ 4 ∈ ℤ
11	1	zlmodzxzel 48843	. . . . 5 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
12	9, 10, 11	mp2an 693	. . . 4 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
13		prelpwi 5394	. . . 4 ⊢ (({⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})) ∧ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))) → {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
14	8, 12, 13	mp2an 693	. . 3 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))
15		eqid 2737	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} = {⟨0, 3⟩, ⟨1, 6⟩}
16		eqid 2737	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} = {⟨0, 2⟩, ⟨1, 4⟩}
17	1, 15, 16	zlmodzxzldep 48992	. . . 4 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})
18	1, 15, 16	ldepsnlinclem1 48993	. . . . . . . 8 ⊢ (𝑓 ∈ ((Base‘ℤ_ring) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}}) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩})
19		simpr 484	. . . . . . . . . . . 12 ⊢ (((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1}))) → ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1})))
20	19	eqcomd 2743	. . . . . . . . . . 11 ⊢ (((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1}))) → (Scalar‘(ℤ_ring freeLMod {0, 1})) = ℤ_ring)
21	2, 20	ax-mp 5	. . . . . . . . . 10 ⊢ (Scalar‘(ℤ_ring freeLMod {0, 1})) = ℤ_ring
22	21	fveq2i 6837	. . . . . . . . 9 ⊢ (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) = (Base‘ℤ_ring)
23	22	oveq1i 7370	. . . . . . . 8 ⊢ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}}) = ((Base‘ℤ_ring) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}})
24	18, 23	eleq2s 2855	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}}) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩})
25	24	a1d 25	. . . . . 6 ⊢ (𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}}) → (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩}))
26	25	rgen 3054	. . . . 5 ⊢ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩})
27	1, 15, 16	ldepsnlinclem2 48994	. . . . . . . 8 ⊢ (𝑓 ∈ ((Base‘ℤ_ring) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}}) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})
28	22	oveq1i 7370	. . . . . . . 8 ⊢ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}}) = ((Base‘ℤ_ring) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}})
29	27, 28	eleq2s 2855	. . . . . . 7 ⊢ (𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}}) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})
30	29	a1d 25	. . . . . 6 ⊢ (𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}}) → (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩}))
31	30	rgen 3054	. . . . 5 ⊢ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})
32		prex 5375	. . . . . 6 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
33		prex 5375	. . . . . 6 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
34		sneq 4578	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → {𝑣} = {{⟨0, 3⟩, ⟨1, 6⟩}})
35	34	difeq2d 4067	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 3⟩, ⟨1, 6⟩}}))
36	1, 15, 16	zlmodzxzldeplem 48986	. . . . . . . . . 10 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}
37		difprsn1 4744	. . . . . . . . . 10 ⊢ ({⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 3⟩, ⟨1, 6⟩}}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
38	36, 37	ax-mp 5	. . . . . . . . 9 ⊢ ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 3⟩, ⟨1, 6⟩}}) = {{⟨0, 2⟩, ⟨1, 4⟩}}
39	35, 38	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
40	39	oveq2d 7376	. . . . . . 7 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) = ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}}))
41	39	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}))
42		id 22	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → 𝑣 = {⟨0, 3⟩, ⟨1, 6⟩})
43	41, 42	neeq12d 2994	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ((𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣 ↔ (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩}))
44	43	imbi2d 340	. . . . . . 7 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ((𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣) ↔ (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩})))
45	40, 44	raleqbidv 3312	. . . . . 6 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → (∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩})))
46		sneq 4578	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → {𝑣} = {{⟨0, 2⟩, ⟨1, 4⟩}})
47	46	difeq2d 4067	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 2⟩, ⟨1, 4⟩}}))
48		difprsn2 4745	. . . . . . . . . 10 ⊢ ({⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 2⟩, ⟨1, 4⟩}}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
49	36, 48	ax-mp 5	. . . . . . . . 9 ⊢ ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {{⟨0, 2⟩, ⟨1, 4⟩}}) = {{⟨0, 3⟩, ⟨1, 6⟩}}
50	47, 49	eqtrdi 2788	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
51	50	oveq2d 7376	. . . . . . 7 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) = ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}}))
52	50	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}))
53		id 22	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → 𝑣 = {⟨0, 2⟩, ⟨1, 4⟩})
54	52, 53	neeq12d 2994	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ((𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣 ↔ (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩}))
55	54	imbi2d 340	. . . . . . 7 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ((𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣) ↔ (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})))
56	51, 55	raleqbidv 3312	. . . . . 6 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → (∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})))
57	32, 33, 45, 56	ralpr 4645	. . . . 5 ⊢ (∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣) ↔ (∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 2⟩, ⟨1, 4⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 2⟩, ⟨1, 4⟩}}) ≠ {⟨0, 3⟩, ⟨1, 6⟩}) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m {{⟨0, 3⟩, ⟨1, 6⟩}})(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1})){{⟨0, 3⟩, ⟨1, 6⟩}}) ≠ {⟨0, 2⟩, ⟨1, 4⟩})))
58	26, 31, 57	mpbir2an 712	. . . 4 ⊢ ∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣)
59	17, 58	pm3.2i 470	. . 3 ⊢ ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣))
60		breq1 5089	. . . . 5 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ↔ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})))
61		id 22	. . . . . 6 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → 𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}})
62		difeq1 4060	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))
63	62	oveq2d 7376	. . . . . . 7 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣})) = ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})))
64	62	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})))
65	64	neeq1d 2992	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → ((𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣 ↔ (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣))
66	65	imbi2d 340	. . . . . . 7 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → ((𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣)))
67	63, 66	raleqbidv 3312	. . . . . 6 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣)))
68	61, 67	raleqbidv 3312	. . . . 5 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣)))
69	60, 68	anbi12d 633	. . . 4 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → ((𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣)) ↔ ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣))))
70	69	rspcev 3565	. . 3 ⊢ (({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})) ∧ ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}}∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣})) ≠ 𝑣))) → ∃𝑠 ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))(𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣)))
71	14, 59, 70	mp2an 693	. 2 ⊢ ∃𝑠 ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))(𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣))
72		fveq2 6834	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘𝑚) = (Base‘(ℤ_ring freeLMod {0, 1})))
73	72	pweqd 4559	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → 𝒫 (Base‘𝑚) = 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
74		breq2 5090	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑠 linDepS 𝑚 ↔ 𝑠 linDepS (ℤ_ring freeLMod {0, 1})))
75		2fveq3 6839	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
76	75	oveq1d 7375	. . . . . . 7 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣})) = ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣})))
77		2fveq3 6839	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (0_g‘(Scalar‘𝑚)) = (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
78	77	breq2d 5098	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑓 finSupp (0_g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1})))))
79		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ( linC ‘𝑚) = ( linC ‘(ℤ_ring freeLMod {0, 1})))
80	79	oveqd 7377	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})))
81	80	neeq1d 2992	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ((𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣 ↔ (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣))
82	78, 81	imbi12d 344	. . . . . . 7 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ((𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ (𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣)))
83	76, 82	raleqbidv 3312	. . . . . 6 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣)))
84	83	ralbidv 3161	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣) ↔ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣)))
85	74, 84	anbi12d 633	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ((𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)) ↔ (𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣))))
86	73, 85	rexeqbidv 3313	. . 3 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)) ↔ ∃𝑠 ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))(𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣))))
87	86	rspcev 3565	. 2 ⊢ (((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ∃𝑠 ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))(𝑠 linDepS (ℤ_ring freeLMod {0, 1}) ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) → (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})) ≠ 𝑣))) → ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣)))
88	3, 71, 87	mp2an 693	1 ⊢ ∃𝑚 ∈ LMod ∃𝑠 ∈ 𝒫 (Base‘𝑚)(𝑠 linDepS 𝑚 ∧ ∀𝑣 ∈ 𝑠 ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑_m (𝑠 ∖ {𝑣}))(𝑓 finSupp (0_g‘(Scalar‘𝑚)) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) ≠ 𝑣))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∖ cdif 3887 𝒫 cpw 4542 {csn 4568 {cpr 4570 ⟨cop 4574 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 finSupp cfsupp 9267 0cc0 11029 1c1 11030 2c2 12227 3c3 12228 4c4 12229 6c6 12231 ℤcz 12515 Basecbs 17170 Scalarcsca 17214 0_gc0g 17393 LModclmod 20846 ℤ_ringczring 21436 freeLMod cfrlm 21736 linC clinc 48892 linDepS clindeps 48929

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 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-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-dvds 16213 df-prm 16632 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-subrng 20514 df-subrg 20538 df-lmod 20848 df-lss 20918 df-sra 21160 df-rgmod 21161 df-cnfld 21345 df-zring 21437 df-dsmm 21722 df-frlm 21737 df-linc 48894 df-lininds 48930 df-lindeps 48932

This theorem is referenced by: ldepslinc 48997

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