ldepsnlinclem1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldepsnlinclem1		Structured version Visualization version GIF version

Theorem ldepsnlinclem1 48498

Description: Lemma 1 for ldepsnlinc 48501. (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
ldepsnlinclem1	⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)

Proof of Theorem ldepsnlinclem1 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 8825	. 2 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → 𝐹:{𝐵}⟶(Base‘ℤ_ring))
2		zlmodzxzldep.b	. . . . 5 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}
3		prex 5395	. . . . 5 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V
4	2, 3	eqeltri 2825	. . . 4 ⊢ 𝐵 ∈ V
5	4	fsn2 7111	. . 3 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) ↔ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}))
6		oveq1 7397	. . . . . 6 ⊢ (𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩} → (𝐹( linC ‘𝑍){𝐵}) = ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}))
7	6	adantl 481	. . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}))
8		zlmodzxzldep.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ_ring freeLMod {0, 1})
9	8	zlmodzxzlmod 48346	. . . . . . . 8 ⊢ (𝑍 ∈ LMod ∧ ℤ_ring = (Scalar‘𝑍))
10	9	simpli 483	. . . . . . 7 ⊢ 𝑍 ∈ LMod
11	10	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝑍 ∈ LMod)
12		2z 12572	. . . . . . . . 9 ⊢ 2 ∈ ℤ
13		4z 12574	. . . . . . . . 9 ⊢ 4 ∈ ℤ
14	8	zlmodzxzel 48347	. . . . . . . . 9 ⊢ ((2 ∈ ℤ ∧ 4 ∈ ℤ) → {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍))
15	12, 13, 14	mp2an 692	. . . . . . . 8 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ (Base‘𝑍)
16	2, 15	eqeltri 2825	. . . . . . 7 ⊢ 𝐵 ∈ (Base‘𝑍)
17	16	a1i 11	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → 𝐵 ∈ (Base‘𝑍))
18		simpl 482	. . . . . 6 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ (Base‘ℤ_ring))
19		eqid 2730	. . . . . . 7 ⊢ (Base‘𝑍) = (Base‘𝑍)
20	9	simpri 485	. . . . . . 7 ⊢ ℤ_ring = (Scalar‘𝑍)
21		eqid 2730	. . . . . . 7 ⊢ (Base‘ℤ_ring) = (Base‘ℤ_ring)
22		eqid 2730	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝑍) = ( ·_𝑠 ‘𝑍)
23	19, 20, 21, 22	lincvalsng 48409	. . . . . 6 ⊢ ((𝑍 ∈ LMod ∧ 𝐵 ∈ (Base‘𝑍) ∧ (𝐹‘𝐵) ∈ (Base‘ℤ_ring)) → ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
24	11, 17, 18, 23	syl3anc 1373	. . . . 5 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ({⟨𝐵, (𝐹‘𝐵)⟩} ( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
25	7, 24	eqtrd 2765	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
26		eqid 2730	. . . . . 6 ⊢ {⟨0, 0⟩, ⟨1, 0⟩} = {⟨0, 0⟩, ⟨1, 0⟩}
27		eqid 2730	. . . . . 6 ⊢ (-_g‘𝑍) = (-_g‘𝑍)
28		zlmodzxzldep.a	. . . . . 6 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}
29	8, 26, 22, 27, 28, 2	zlmodzxznm 48490	. . . . 5 ⊢ ∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
30		r19.26 3092	. . . . . 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))
31		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑖 = (𝐹‘𝐵) → (𝑖( ·_𝑠 ‘𝑍)𝐵) = ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵))
32	31	neeq1d 2985	. . . . . . . . 9 ⊢ (𝑖 = (𝐹‘𝐵) → ((𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 ↔ ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
33	32	rspcv 3587	. . . . . . . 8 ⊢ ((𝐹‘𝐵) ∈ ℤ → (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
34		zringbas 21370	. . . . . . . . . . . 12 ⊢ ℤ = (Base‘ℤ_ring)
35	34	eqcomi 2739	. . . . . . . . . . 11 ⊢ (Base‘ℤ_ring) = ℤ
36	35	eleq2i 2821	. . . . . . . . . 10 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ↔ (𝐹‘𝐵) ∈ ℤ)
37	36	biimpi 216	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (Base‘ℤ_ring) → (𝐹‘𝐵) ∈ ℤ)
38	37	adantr 480	. . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹‘𝐵) ∈ ℤ)
39	33, 38	syl11 33	. . . . . . 7 ⊢ (∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴 → (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
40	39	adantl 481	. . . . . 6 ⊢ ((∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ ∀𝑖 ∈ ℤ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
41	30, 40	sylbi 217	. . . . 5 ⊢ (∀𝑖 ∈ ℤ ((𝑖( ·_𝑠 ‘𝑍)𝐴) ≠ 𝐵 ∧ (𝑖( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴) → (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴))
42	29, 41	ax-mp 5	. . . 4 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → ((𝐹‘𝐵)( ·_𝑠 ‘𝑍)𝐵) ≠ 𝐴)
43	25, 42	eqnetrd 2993	. . 3 ⊢ (((𝐹‘𝐵) ∈ (Base‘ℤ_ring) ∧ 𝐹 = {⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
44	5, 43	sylbi 217	. 2 ⊢ (𝐹:{𝐵}⟶(Base‘ℤ_ring) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)
45	1, 44	syl 17	1 ⊢ (𝐹 ∈ ((Base‘ℤ_ring) ↑_m {𝐵}) → (𝐹( linC ‘𝑍){𝐵}) ≠ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∀wral 3045 Vcvv 3450 {csn 4592 {cpr 4594 ⟨cop 4598 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 0cc0 11075 1c1 11076 2c2 12248 3c3 12249 4c4 12250 6c6 12252 ℤcz 12536 Basecbs 17186 Scalarcsca 17230 ·_𝑠 cvsca 17231 -_gcsg 18874 LModclmod 20773 ℤ_ringczring 21363 freeLMod cfrlm 21662 linC clinc 48397

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-prm 16649 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-subrng 20462 df-subrg 20486 df-lmod 20775 df-lss 20845 df-sra 21087 df-rgmod 21088 df-cnfld 21272 df-zring 21364 df-dsmm 21648 df-frlm 21663 df-linc 48399

This theorem is referenced by: ldepsnlinc 48501

W3C validator