ldepsnlinc - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinc 49951

Description: The reverse implication of islindeps2 49926 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 49939 and zlmodzxznm 49940. (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 2763	. . . 4 ⊢ (ℤ_ring freeLMod {0, 1}) = (ℤ_ring freeLMod {0, 1})
2	1	zlmodzxzlmod 49797	. . 3 ⊢ ((ℤ_ring freeLMod {0, 1}) ∈ LMod ∧ ℤ_ring = (Scalar‘(ℤ_ring freeLMod {0, 1})))
3	2	simpli 489	. 2 ⊢ (ℤ_ring freeLMod {0, 1}) ∈ LMod
4		3z 12769	. . . . 5 ⊢ 3 ∈ ℤ
5		6nn 12472	. . . . . 6 ⊢ 6 ∈ ℕ
6	5	nnzi 12760	. . . . 5 ⊢ 6 ∈ ℤ
7	1	zlmodzxzel 49798	. . . . 5 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
8	4, 6, 7	mp2an 705	. . . 4 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
9		2z 12768	. . . . 5 ⊢ 2 ∈ ℤ
10		4z 12770	. . . . 5 ⊢ 4 ∈ ℤ
11	1	zlmodzxzel 49798	. . . . 5 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1})))
12	9, 10, 11	mp2an 705	. . . 4 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘(ℤ_ring freeLMod {0, 1}))
13		prelpwi 5422	. . . 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 705	. . 3 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∈ 𝒫 (Base‘(ℤ_ring freeLMod {0, 1}))
15		eqid 2763	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} = {⟨0, 3⟩, ⟨1, 6⟩}
16		eqid 2763	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} = {⟨0, 2⟩, ⟨1, 4⟩}
17	1, 15, 16	zlmodzxzldep 49947	. . . 4 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})
18	1, 15, 16	ldepsnlinclem1 49948	. . . . . . . 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 490	. . . . . . . . . . . 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 2769	. . . . . . . . . . 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 6919	. . . . . . . . 9 ⊢ (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) = (Base‘ℤ_ring)
23	22	oveq1i 7461	. . . . . . . 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 2882	. . . . . . 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 26	. . . . . 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 3082	. . . . 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 49949	. . . . . . . 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 7461	. . . . . . . 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 2882	. . . . . . 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 26	. . . . . 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 3082	. . . . 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 5403	. . . . . 6 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
33		prex 5403	. . . . . 6 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
34		sneq 4594	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → {𝑣} = {{⟨0, 3⟩, ⟨1, 6⟩}})
35	34	difeq2d 4066	. . . . . . . . 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 49941	. . . . . . . . . 10 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}
37		difprsn1 4765	. . . . . . . . . 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 2814	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
40	39	oveq2d 7467	. . . . . . 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 7467	. . . . . . . . 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 23	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → 𝑣 = {⟨0, 3⟩, ⟨1, 6⟩})
43	41, 42	neeq12d 3020	. . . . . . . 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 343	. . . . . . 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 3338	. . . . . 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 4594	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → {𝑣} = {{⟨0, 2⟩, ⟨1, 4⟩}})
47	46	difeq2d 4066	. . . . . . . . 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 4766	. . . . . . . . . 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 2814	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
51	50	oveq2d 7467	. . . . . . 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 7467	. . . . . . . . 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 23	. . . . . . . . 9 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → 𝑣 = {⟨0, 2⟩, ⟨1, 4⟩})
54	52, 53	neeq12d 3020	. . . . . . . 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 343	. . . . . . 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 3338	. . . . . 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 4662	. . . . 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 724	. . . 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 476	. . 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 5111	. . . . 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 23	. . . . . 6 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → 𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}})
62		difeq1 4059	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))
63	62	oveq2d 7467	. . . . . . 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 7467	. . . . . . . . 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 3018	. . . . . . . 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 343	. . . . . . 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 3338	. . . . . 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 3338	. . . . 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 644	. . . 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 3580	. . 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 705	. 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 6916	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘𝑚) = (Base‘(ℤ_ring freeLMod {0, 1})))
73	72	pweqd 4577	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → 𝒫 (Base‘𝑚) = 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
74		breq2 5112	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑠 linDepS 𝑚 ↔ 𝑠 linDepS (ℤ_ring freeLMod {0, 1})))
75		2fveq3 6921	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
76	75	oveq1d 7466	. . . . . . 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 6921	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (0_g‘(Scalar‘𝑚)) = (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
78	77	breq2d 5120	. . . . . . . 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 6916	. . . . . . . . . 10 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ( linC ‘𝑚) = ( linC ‘(ℤ_ring freeLMod {0, 1})))
80	79	oveqd 7468	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})))
81	80	neeq1d 3018	. . . . . . . 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 347	. . . . . . 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 3338	. . . . . 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 3189	. . . . 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 644	. . . 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 3339	. . 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 3580	. 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 705	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 ∖ cdif 4048 𝒫 cpw 4560 {csn 4588 {cpr 4590 ⟨cop 4840 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ↑_m cmap 8885 finSupp cfsupp 9391 0cc0 11239 1c1 11240 2c2 12437 3c3 12438 4c4 12439 6c6 12441 ℤcz 12733 Basecbs 17431 Scalarcsca 17475 0_gc0g 17660 LModclmod 21186 ℤ_ringczring 21806 freeLMod cfrlm 22107 linC clinc 49847 linDepS clindeps 49884

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318 ax-mulf 11319

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-sup 9471 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467 df-prm 16891 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-0g 17662 df-gsum 17663 df-prds 17668 df-pws 17670 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-sbg 19199 df-mulg 19328 df-subg 19383 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-subrng 20849 df-subrg 20873 df-lmod 21188 df-lss 21258 df-sra 21499 df-rgmod 21500 df-cnfld 21730 df-zring 21807 df-dsmm 22093 df-frlm 22108 df-linc 49849 df-lininds 49885 df-lindeps 49887

This theorem is used by: ldepslinc 49952

W3C validator