ldepsnlinc - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinc 48451

Description: The reverse implication of islindeps2 48426 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 48439 and zlmodzxznm 48440. (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 2736	. . . 4 ⊢ (ℤ_ring freeLMod {0, 1}) = (ℤ_ring freeLMod {0, 1})
2	1	zlmodzxzlmod 48296	. . 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 12630	. . . . 5 ⊢ 3 ∈ ℤ
5		6nn 12334	. . . . . 6 ⊢ 6 ∈ ℕ
6	5	nnzi 12621	. . . . 5 ⊢ 6 ∈ ℤ
7	1	zlmodzxzel 48297	. . . . 5 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
8	4, 6, 7	mp2an 692	. . . 4 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
9		2z 12629	. . . . 5 ⊢ 2 ∈ ℤ
10		4z 12631	. . . . 5 ⊢ 4 ∈ ℤ
11	1	zlmodzxzel 48297	. . . . 5 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
12	9, 10, 11	mp2an 692	. . . 4 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
13		prelpwi 5427	. . . 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 692	. . 3 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))
15		eqid 2736	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} = {⟨0, 3⟩, ⟨1, 6⟩}
16		eqid 2736	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} = {⟨0, 2⟩, ⟨1, 4⟩}
17	1, 15, 16	zlmodzxzldep 48447	. . . 4 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})
18	1, 15, 16	ldepsnlinclem1 48448	. . . . . . . 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 2742	. . . . . . . . . . 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 6884	. . . . . . . . 9 ⊢ (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) = (Base‘ℤ_ring)
23	22	oveq1i 7420	. . . . . . . 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 2853	. . . . . . 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 48449	. . . . . . . 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 7420	. . . . . . . 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 2853	. . . . . . 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 5412	. . . . . 6 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
33		prex 5412	. . . . . 6 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
34		sneq 4616	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → {𝑣} = {{⟨0, 3⟩, ⟨1, 6⟩}})
35	34	difeq2d 4106	. . . . . . . . 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 48441	. . . . . . . . . 10 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}
37		difprsn1 4781	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
40	39	oveq2d 7426	. . . . . . 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 7426	. . . . . . . . 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 3329	. . . . . 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 4616	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → {𝑣} = {{⟨0, 2⟩, ⟨1, 4⟩}})
47	46	difeq2d 4106	. . . . . . . . 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 4782	. . . . . . . . . 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 2787	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
51	50	oveq2d 7426	. . . . . . 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 7426	. . . . . . . . 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 3329	. . . . . 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 4681	. . . . 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 711	. . . 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 5127	. . . . 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 4099	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))
63	62	oveq2d 7426	. . . . . . 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 7426	. . . . . . . . 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 3329	. . . . . 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 3329	. . . . 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 632	. . . 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 3606	. . 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 692	. 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 6881	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘𝑚) = (Base‘(ℤ_ring freeLMod {0, 1})))
73	72	pweqd 4597	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → 𝒫 (Base‘𝑚) = 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
74		breq2 5128	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑠 linDepS 𝑚 ↔ 𝑠 linDepS (ℤ_ring freeLMod {0, 1})))
75		2fveq3 6886	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
76	75	oveq1d 7425	. . . . . . 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 6886	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (0_g‘(Scalar‘𝑚)) = (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
78	77	breq2d 5136	. . . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ( linC ‘𝑚) = ( linC ‘(ℤ_ring freeLMod {0, 1})))
80	79	oveqd 7427	. . . . . . . . 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 3329	. . . . . 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 3164	. . . . 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 632	. . . 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 3330	. . 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 3606	. 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 692	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 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∃wrex 3061 ∖ cdif 3928 𝒫 cpw 4580 {csn 4606 {cpr 4608 ⟨cop 4612 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 finSupp cfsupp 9378 0cc0 11134 1c1 11135 2c2 12300 3c3 12301 4c4 12302 6c6 12304 ℤcz 12593 Basecbs 17233 Scalarcsca 17279 0_gc0g 17458 LModclmod 20822 ℤ_ringczring 21412 freeLMod cfrlm 21711 linC clinc 48347 linDepS clindeps 48384

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-sup 9459 df-oi 9529 df-card 9958 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-rp 13014 df-fz 13530 df-fzo 13677 df-seq 14025 df-exp 14085 df-hash 14354 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-prm 16696 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-0g 17460 df-gsum 17461 df-prds 17466 df-pws 17468 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-cntz 19305 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-subrng 20511 df-subrg 20535 df-lmod 20824 df-lss 20894 df-sra 21136 df-rgmod 21137 df-cnfld 21321 df-zring 21413 df-dsmm 21697 df-frlm 21712 df-linc 48349 df-lininds 48385 df-lindeps 48387

This theorem is referenced by: ldepslinc 48452

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