ldepsnlinclem2 - Mathbox for Alexander van der Vekens

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Theorem ldepsnlinclem2 48352

Description: Lemma 2 for ldepsnlinc 48354. (Contributed by AV, 25-May-2019.) (Revised by AV, 10-Jun-2019.)

**Hypotheses**
Ref	Expression
zlmodzxzldep.z	⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
zlmodzxzldep.a	⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
zlmodzxzldep.b	⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}

**Assertion**
Ref	Expression
ldepsnlinclem2	⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)

Proof of Theorem ldepsnlinclem2 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8888	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → 𝐹:{𝐴}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.a	. . . . 5 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
3		prex 5443	. . . . 5 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V
4	2, 3	eqeltri 2835	. . . 4 ⊢ 𝐴 ∈ V
5	4	fsn2 7156	. . 3 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}))
6		oveq1 7438	. . . . . 6 ⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} → (𝐹( linC ‘𝑍){𝐴}) = ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}))
7	6	adantl 481	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}))
8		zlmodzxzldep.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
9	8	zlmodzxzlmod 48199	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 483	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝑍 ∈ LMod)
12		3z 12648	. . . . . . . . 9 ⊢ 3 ∈ ℤ
13		6nn 12353	. . . . . . . . . 10 ⊢ 6 ∈ ℕ
14	13	nnzi 12639	. . . . . . . . 9 ⊢ 6 ∈ ℤ
15	8	zlmodzxzel 48200	. . . . . . . . 9 ⊢ ((3 ∈ ℤ ∧ 6 ∈ ℤ) → {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍))
16	12, 14, 15	mp2an 692	. . . . . . . 8 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ (Base‘𝑍)
17	2, 16	eqeltri 2835	. . . . . . 7 ⊢ 𝐴 ∈ (Base‘𝑍)
18	17	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → 𝐴 ∈ (Base‘𝑍))
19		simpl 482	. . . . . 6 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹‘𝐴) ∈ (Base‘ℤ_ring))
20		eqid 2735	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
21	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
22		eqid 2735	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
23		eqid 2735	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
24	20, 21, 22, 23	lincvalsng 48262	. . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐴 ∈ (Base‘𝑍) ∧ (𝐹‘𝐴) ∈ (Base‘ℤ_ring)) → ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
25	11, 18, 19, 24	syl3anc 1370	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ({⟨𝐴, (𝐹‘𝐴)⟩} ( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
26	7, 25	eqtrd 2775	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
27		eqid 2735	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
28		eqid 2735	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
29		zlmodzxzldep.b	. . . . . 6 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
30	8, 27, 23, 28, 2, 29	zlmodzxznm 48343	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
31		r19.26 3109	. . . . . 6 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) ↔ (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
32		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐴) → (𝑖( ·_𝑠 ‘𝑍)𝐴) = ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴))
33	32	neeq1d 2998	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐴) → ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ↔ ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
34	33	rspcv 3618	. . . . . . . 8 ⊢ ((𝐹‘𝐴) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
35		zringbas 21482	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
36	35	eqcomi 2744	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
37	36	eleq2i 2831	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐴) ∈ ℤ)
38	37	biimpi 216	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ∈ (Base‘ℤ_ring) → (𝐹‘𝐴) ∈ ℤ)
39	38	adantr 480	. . . . . . . 8 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹‘𝐴) ∈ ℤ)
40	34, 39	syl11 33	. . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
41	40	adantr 480	. . . . . 6 ⊢ ((∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
42	31, 41	sylbi 217	. . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵))
43	30, 42	ax-mp 5	. . . 4 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → ((𝐹‘𝐴)( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵)
44	26, 43	eqnetrd 3006	. . 3 ⊢ (((𝐹‘𝐴) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
45	5, 44	sylbi 217	. 2 ⊢ (𝐹:{𝐴}⟶(Base‘ℤ_ring) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)
46	1, 45	syl 17	1 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐴}) → (𝐹( linC ‘𝑍){𝐴}) ≠ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 Vcvv 3478 {csn 4631 {cpr 4633 ⟨cop 4637 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 0cc0 11153 1c1 11154 2c2 12319 3c3 12320 4c4 12321 6c6 12323 ℤcz 12611 Basecbs 17245 Scalarcsca 17301 ·_𝑠 cvsca 17302 -_gcsg 18966 LModclmod 20875 ℤ_ringczring 21475 freeLMod cfrlm 21784 linC clinc 48250

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-prm 16706 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-cntz 19348 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-dsmm 21770 df-frlm 21785 df-linc 48252

This theorem is referenced by: ldepsnlinc 48354

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