![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for zlmodzxzldep 43141. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.) |
Ref | Expression |
---|---|
zlmodzxzldep.z | ⊢ 𝑍 = (ℤring freeLMod {0, 1}) |
zlmodzxzldep.a | ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} |
zlmodzxzldep.b | ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} |
zlmodzxzldeplem.f | ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} |
Ref | Expression |
---|---|
zlmodzxzldeplem4 | ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmodzxzldep.a | . . 3 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
2 | prex 5131 | . . 3 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
3 | 1, 2 | eqeltri 2903 | . 2 ⊢ 𝐴 ∈ V |
4 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
5 | prex 5131 | . . 3 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
6 | 4, 5 | eqeltri 2903 | . 2 ⊢ 𝐵 ∈ V |
7 | 2ne0 11463 | . . . . 5 ⊢ 2 ≠ 0 | |
8 | zlmodzxzldeplem.f | . . . . . . . 8 ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} | |
9 | 8 | fveq1i 6435 | . . . . . . 7 ⊢ (𝐹‘𝐴) = ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝐴) |
10 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
11 | 10, 1, 4 | zlmodzxzldeplem 43135 | . . . . . . . 8 ⊢ 𝐴 ≠ 𝐵 |
12 | 2ex 11429 | . . . . . . . . 9 ⊢ 2 ∈ V | |
13 | 3, 12 | fvpr1 6713 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝐴) = 2) |
14 | 11, 13 | mp1i 13 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝐴) = 2) |
15 | 9, 14 | syl5eq 2874 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) = 2) |
16 | 15 | neeq1d 3059 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ↔ 2 ≠ 0)) |
17 | 7, 16 | mpbiri 250 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) ≠ 0) |
18 | 17 | orcd 906 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0)) |
19 | fveq2 6434 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
20 | 19 | neeq1d 3059 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐴) ≠ 0)) |
21 | fveq2 6434 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
22 | 21 | neeq1d 3059 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐵) ≠ 0)) |
23 | 20, 22 | rexprg 4455 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 ↔ ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0))) |
24 | 18, 23 | mpbird 249 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0) |
25 | 3, 6, 24 | mp2an 685 | 1 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 ∃wrex 3119 Vcvv 3415 {cpr 4400 〈cop 4404 ‘cfv 6124 (class class class)co 6906 0cc0 10253 1c1 10254 -cneg 10587 2c2 11407 3c3 11408 4c4 11409 6c6 11411 ℤringzring 20179 freeLMod cfrlm 20454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-2 11415 df-3 11416 |
This theorem is referenced by: zlmodzxzldep 43141 |
Copyright terms: Public domain | W3C validator |