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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for zlmodzxzldep 44553. (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 5324 | . . 3 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
3 | 1, 2 | eqeltri 2909 | . 2 ⊢ 𝐴 ∈ V |
4 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
5 | prex 5324 | . . 3 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
6 | 4, 5 | eqeltri 2909 | . 2 ⊢ 𝐵 ∈ V |
7 | 2ne0 11735 | . . . . 5 ⊢ 2 ≠ 0 | |
8 | zlmodzxzldeplem.f | . . . . . . . 8 ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} | |
9 | 8 | fveq1i 6665 | . . . . . . 7 ⊢ (𝐹‘𝐴) = ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝐴) |
10 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
11 | 10, 1, 4 | zlmodzxzldeplem 44547 | . . . . . . . 8 ⊢ 𝐴 ≠ 𝐵 |
12 | 2ex 11708 | . . . . . . . . 9 ⊢ 2 ∈ V | |
13 | 3, 12 | fvpr1 6946 | . . . . . . . 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 2868 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) = 2) |
16 | 15 | neeq1d 3075 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ↔ 2 ≠ 0)) |
17 | 7, 16 | mpbiri 260 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) ≠ 0) |
18 | 17 | orcd 869 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0)) |
19 | fveq2 6664 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
20 | 19 | neeq1d 3075 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐴) ≠ 0)) |
21 | fveq2 6664 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
22 | 21 | neeq1d 3075 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐵) ≠ 0)) |
23 | 20, 22 | rexprg 4626 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 ↔ ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0))) |
24 | 18, 23 | mpbird 259 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0) |
25 | 3, 6, 24 | mp2an 690 | 1 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 Vcvv 3494 {cpr 4562 〈cop 4566 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 -cneg 10865 2c2 11686 3c3 11687 4c4 11688 6c6 11690 ℤringzring 20611 freeLMod cfrlm 20884 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-2 11694 df-3 11695 |
This theorem is referenced by: zlmodzxzldep 44553 |
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