ldepsnlinc - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinc 47276

Description: The reverse implication of islindeps2 47251 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 combinantion of the other elements of this subset. Such a left module can be constructed by using zlmodzxzequa 47264 and zlmodzxznm 47265. (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 2730	. . . 4 ⊢ (ℤ_ring freeLMod {0, 1}) = (ℤ_ring freeLMod {0, 1})
2	1	zlmodzxzlmod 47118	. . 3 ⊢ ((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1})))
3	2	simpli 482	. 2 ⊢ (ℤ_ring freeLMod {0, 1}) ∈ LMod
4		3z 12599	. . . . 5 ⊢ 3 ∈ ℤ
5		6nn 12305	. . . . . 6 ⊢ 6 ∈ ℕ
6	5	nnzi 12590	. . . . 5 ⊢ 6 ∈ ℤ
7	1	zlmodzxzel 47119	. . . . 5 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
8	4, 6, 7	mp2an 688	. . . 4 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
9		2z 12598	. . . . 5 ⊢ 2 ∈ ℤ
10		4z 12600	. . . . 5 ⊢ 4 ∈ ℤ
11	1	zlmodzxzel 47119	. . . . 5 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
12	9, 10, 11	mp2an 688	. . . 4 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
13		prelpwi 5446	. . . 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 688	. . 3 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))
15		eqid 2730	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} = {⟨0, 3⟩, ⟨1, 6⟩}
16		eqid 2730	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} = {⟨0, 2⟩, ⟨1, 4⟩}
17	1, 15, 16	zlmodzxzldep 47272	. . . 4 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})
18	1, 15, 16	ldepsnlinclem1 47273	. . . . . . . 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 483	. . . . . . . . . . . 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 2736	. . . . . . . . . . 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 6893	. . . . . . . . 9 ⊢ (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) = (Base‘ℤ_ring)
23	22	oveq1i 7421	. . . . . . . 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 2849	. . . . . . 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 3061	. . . . 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 47274	. . . . . . . 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 7421	. . . . . . . 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 2849	. . . . . . 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 3061	. . . . 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 5431	. . . . . 6 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
33		prex 5431	. . . . . 6 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
34		sneq 4637	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → {𝑣} = {{⟨0, 3⟩, ⟨1, 6⟩}})
35	34	difeq2d 4121	. . . . . . . . 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 47266	. . . . . . . . . 10 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}
37		difprsn1 4802	. . . . . . . . . 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 2786	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
40	39	oveq2d 7427	. . . . . . 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 7427	. . . . . . . . 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 3000	. . . . . . . 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 339	. . . . . . 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 3340	. . . . . 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 4637	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → {𝑣} = {{⟨0, 2⟩, ⟨1, 4⟩}})
47	46	difeq2d 4121	. . . . . . . . 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 4803	. . . . . . . . . 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 2786	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
51	50	oveq2d 7427	. . . . . . 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 7427	. . . . . . . . 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 3000	. . . . . . . 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 339	. . . . . . 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 3340	. . . . . 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 4703	. . . . 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 707	. . . 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 469	. . 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 5150	. . . . 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 4114	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))
63	62	oveq2d 7427	. . . . . . 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 7427	. . . . . . . . 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 2998	. . . . . . . 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 339	. . . . . . 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 3340	. . . . . 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 3340	. . . . 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 629	. . . 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 3611	. . 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 688	. 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 6890	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘𝑚) = (Base‘(ℤ_ring freeLMod {0, 1})))
73	72	pweqd 4618	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → 𝒫 (Base‘𝑚) = 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
74		breq2 5151	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑠 linDepS 𝑚 ↔ 𝑠 linDepS (ℤ_ring freeLMod {0, 1})))
75		2fveq3 6895	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
76	75	oveq1d 7426	. . . . . . 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 6895	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (0_g‘(Scalar‘𝑚)) = (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
78	77	breq2d 5159	. . . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ( linC ‘𝑚) = ( linC ‘(ℤ_ring freeLMod {0, 1})))
80	79	oveqd 7428	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})))
81	80	neeq1d 2998	. . . . . . . 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 343	. . . . . . 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 3340	. . . . . 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 3175	. . . . 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 629	. . . 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 3341	. . 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 3611	. 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 688	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 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∖ cdif 3944 𝒫 cpw 4601 {csn 4627 {cpr 4629 ⟨cop 4633 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 finSupp cfsupp 9363 0cc0 11112 1c1 11113 2c2 12271 3c3 12272 4c4 12273 6c6 12275 ℤcz 12562 Basecbs 17148 Scalarcsca 17204 0_gc0g 17389 LModclmod 20614 ℤ_ringczring 21217 freeLMod cfrlm 21520 linC clinc 47172 linDepS clindeps 47209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-dvds 16202 df-prm 16613 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-subrng 20434 df-subrg 20459 df-lmod 20616 df-lss 20687 df-sra 20930 df-rgmod 20931 df-cnfld 21145 df-zring 21218 df-dsmm 21506 df-frlm 21521 df-linc 47174 df-lininds 47210 df-lindeps 47212

This theorem is referenced by: ldepslinc 47277

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