ldepsnlinc - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinc 48425

Description: The reverse implication of islindeps2 48400 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 48413 and zlmodzxznm 48414. (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 48270	. . 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 12650	. . . . 5 ⊢ 3 ∈ ℤ
5		6nn 12355	. . . . . 6 ⊢ 6 ∈ ℕ
6	5	nnzi 12641	. . . . 5 ⊢ 6 ∈ ℤ
7	1	zlmodzxzel 48271	. . . . 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 12649	. . . . 5 ⊢ 2 ∈ ℤ
10		4z 12651	. . . . 5 ⊢ 4 ∈ ℤ
11	1	zlmodzxzel 48271	. . . . 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 5452	. . . 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 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 48421	. . . 4 ⊢ {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} linDepS (ℤ_ring freeLMod {0, 1})
18	1, 15, 16	ldepsnlinclem1 48422	. . . . . . . 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 6909	. . . . . . . . 9 ⊢ (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))) = (Base‘ℤ_ring)
23	22	oveq1i 7441	. . . . . . . 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 2859	. . . . . . 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 3063	. . . . 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 48423	. . . . . . . 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 7441	. . . . . . . 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 2859	. . . . . . 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 3063	. . . . 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 5437	. . . . . 6 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
33		prex 5437	. . . . . 6 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
34		sneq 4636	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → {𝑣} = {{⟨0, 3⟩, ⟨1, 6⟩}})
35	34	difeq2d 4126	. . . . . . . . 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 48415	. . . . . . . . . 10 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}
37		difprsn1 4800	. . . . . . . . . 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 2793	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 3⟩, ⟨1, 6⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 2⟩, ⟨1, 4⟩}})
40	39	oveq2d 7447	. . . . . . 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 7447	. . . . . . . . 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 3002	. . . . . . . 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 3346	. . . . . 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 4636	. . . . . . . . . 10 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → {𝑣} = {{⟨0, 2⟩, ⟨1, 4⟩}})
47	46	difeq2d 4126	. . . . . . . . 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 4801	. . . . . . . . . 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 2793	. . . . . . . 8 ⊢ (𝑣 = {⟨0, 2⟩, ⟨1, 4⟩} → ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}) = {{⟨0, 3⟩, ⟨1, 6⟩}})
51	50	oveq2d 7447	. . . . . . 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 7447	. . . . . . . . 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 3002	. . . . . . . 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 3346	. . . . . 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 4700	. . . . 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 5146	. . . . 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 4119	. . . . . . . 8 ⊢ (𝑠 = {{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} → (𝑠 ∖ {𝑣}) = ({{⟨0, 3⟩, ⟨1, 6⟩}, {⟨0, 2⟩, ⟨1, 4⟩}} ∖ {𝑣}))
63	62	oveq2d 7447	. . . . . . 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 7447	. . . . . . . . 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 3000	. . . . . . . 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 3346	. . . . . 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 3346	. . . . 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 3622	. . 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 6906	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘𝑚) = (Base‘(ℤ_ring freeLMod {0, 1})))
73	72	pweqd 4617	. . . 4 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → 𝒫 (Base‘𝑚) = 𝒫 (Base‘(ℤ_ring freeLMod {0, 1})))
74		breq2 5147	. . . . 5 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑠 linDepS 𝑚 ↔ 𝑠 linDepS (ℤ_ring freeLMod {0, 1})))
75		2fveq3 6911	. . . . . . . 8 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (Base‘(Scalar‘𝑚)) = (Base‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
76	75	oveq1d 7446	. . . . . . 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 6911	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (0_g‘(Scalar‘𝑚)) = (0_g‘(Scalar‘(ℤ_ring freeLMod {0, 1}))))
78	77	breq2d 5155	. . . . . . . 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 6906	. . . . . . . . . 10 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → ( linC ‘𝑚) = ( linC ‘(ℤ_ring freeLMod {0, 1})))
80	79	oveqd 7448	. . . . . . . . 9 ⊢ (𝑚 = (ℤ_ring freeLMod {0, 1}) → (𝑓( linC ‘𝑚)(𝑠 ∖ {𝑣})) = (𝑓( linC ‘(ℤ_ring freeLMod {0, 1}))(𝑠 ∖ {𝑣})))
81	80	neeq1d 3000	. . . . . . . 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 3346	. . . . . 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 3178	. . . . 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 3347	. . 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 3622	. 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 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 𝒫 cpw 4600 {csn 4626 {cpr 4628 ⟨cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ↑_m cmap 8866 finSupp cfsupp 9401 0cc0 11155 1c1 11156 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤcz 12613 Basecbs 17247 Scalarcsca 17300 0_gc0g 17484 LModclmod 20858 ℤ_ringczring 21457 freeLMod cfrlm 21766 linC clinc 48321 linDepS clindeps 48358

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-subrng 20546 df-subrg 20570 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-zring 21458 df-dsmm 21752 df-frlm 21767 df-linc 48323 df-lininds 48359 df-lindeps 48361

This theorem is referenced by: ldepslinc 48426

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