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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for zlmodzxzldep 45797. (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 5358 | . . 3 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
3 | 1, 2 | eqeltri 2836 | . 2 ⊢ 𝐴 ∈ V |
4 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
5 | prex 5358 | . . 3 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
6 | 4, 5 | eqeltri 2836 | . 2 ⊢ 𝐵 ∈ V |
7 | 2ne0 12060 | . . . . 5 ⊢ 2 ≠ 0 | |
8 | zlmodzxzldeplem.f | . . . . . . . 8 ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} | |
9 | 8 | fveq1i 6769 | . . . . . . 7 ⊢ (𝐹‘𝐴) = ({〈𝐴, 2〉, 〈𝐵, -3〉}‘𝐴) |
10 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
11 | 10, 1, 4 | zlmodzxzldeplem 45791 | . . . . . . . 8 ⊢ 𝐴 ≠ 𝐵 |
12 | 2ex 12033 | . . . . . . . . 9 ⊢ 2 ∈ V | |
13 | 3, 12 | fvpr1 7059 | . . . . . . . 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 | eqtrid 2791 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) = 2) |
16 | 15 | neeq1d 3004 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ↔ 2 ≠ 0)) |
17 | 7, 16 | mpbiri 257 | . . . 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 6768 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
20 | 19 | neeq1d 3004 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐴) ≠ 0)) |
21 | fveq2 6768 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
22 | 21 | neeq1d 3004 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐵) ≠ 0)) |
23 | 20, 22 | rexprg 4637 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 ↔ ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0))) |
24 | 18, 23 | mpbird 256 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0) |
25 | 3, 6, 24 | mp2an 688 | 1 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 843 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∃wrex 3066 Vcvv 3430 {cpr 4568 〈cop 4572 ‘cfv 6430 (class class class)co 7268 0cc0 10855 1c1 10856 -cneg 11189 2c2 12011 3c3 12012 4c4 12013 6c6 12015 ℤringczring 20651 freeLMod cfrlm 20934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-2 12019 df-3 12020 |
This theorem is referenced by: zlmodzxzldep 45797 |
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