![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem | Structured version Visualization version GIF version |
Description: A and B are not equal. (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⟩} |
Ref | Expression |
---|---|
zlmodzxzldeplem | ⊢ 𝐴 ≠ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5454 | . . . . 5 ⊢ ⟨0, 3⟩ ∈ V | |
2 | opex 5454 | . . . . 5 ⊢ ⟨1, 6⟩ ∈ V | |
3 | 1, 2 | pm3.2i 470 | . . . 4 ⊢ (⟨0, 3⟩ ∈ V ∧ ⟨1, 6⟩ ∈ V) |
4 | opex 5454 | . . . . 5 ⊢ ⟨0, 2⟩ ∈ V | |
5 | opex 5454 | . . . . 5 ⊢ ⟨1, 4⟩ ∈ V | |
6 | 4, 5 | pm3.2i 470 | . . . 4 ⊢ (⟨0, 2⟩ ∈ V ∧ ⟨1, 4⟩ ∈ V) |
7 | 3, 6 | pm3.2i 470 | . . 3 ⊢ ((⟨0, 3⟩ ∈ V ∧ ⟨1, 6⟩ ∈ V) ∧ (⟨0, 2⟩ ∈ V ∧ ⟨1, 4⟩ ∈ V)) |
8 | 2re 12283 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 2lt3 12381 | . . . . . . . 8 ⊢ 2 < 3 | |
10 | 8, 9 | gtneii 11323 | . . . . . . 7 ⊢ 3 ≠ 2 |
11 | 10 | olci 863 | . . . . . 6 ⊢ (0 ≠ 0 ∨ 3 ≠ 2) |
12 | c0ex 11205 | . . . . . . 7 ⊢ 0 ∈ V | |
13 | 3ex 12291 | . . . . . . 7 ⊢ 3 ∈ V | |
14 | 12, 13 | opthne 5472 | . . . . . 6 ⊢ (⟨0, 3⟩ ≠ ⟨0, 2⟩ ↔ (0 ≠ 0 ∨ 3 ≠ 2)) |
15 | 11, 14 | mpbir 230 | . . . . 5 ⊢ ⟨0, 3⟩ ≠ ⟨0, 2⟩ |
16 | 0ne1 12280 | . . . . . . 7 ⊢ 0 ≠ 1 | |
17 | 16 | orci 862 | . . . . . 6 ⊢ (0 ≠ 1 ∨ 3 ≠ 4) |
18 | 12, 13 | opthne 5472 | . . . . . 6 ⊢ (⟨0, 3⟩ ≠ ⟨1, 4⟩ ↔ (0 ≠ 1 ∨ 3 ≠ 4)) |
19 | 17, 18 | mpbir 230 | . . . . 5 ⊢ ⟨0, 3⟩ ≠ ⟨1, 4⟩ |
20 | 15, 19 | pm3.2i 470 | . . . 4 ⊢ (⟨0, 3⟩ ≠ ⟨0, 2⟩ ∧ ⟨0, 3⟩ ≠ ⟨1, 4⟩) |
21 | 20 | orci 862 | . . 3 ⊢ ((⟨0, 3⟩ ≠ ⟨0, 2⟩ ∧ ⟨0, 3⟩ ≠ ⟨1, 4⟩) ∨ (⟨1, 6⟩ ≠ ⟨0, 2⟩ ∧ ⟨1, 6⟩ ≠ ⟨1, 4⟩)) |
22 | prneimg 4847 | . . 3 ⊢ (((⟨0, 3⟩ ∈ V ∧ ⟨1, 6⟩ ∈ V) ∧ (⟨0, 2⟩ ∈ V ∧ ⟨1, 4⟩ ∈ V)) → (((⟨0, 3⟩ ≠ ⟨0, 2⟩ ∧ ⟨0, 3⟩ ≠ ⟨1, 4⟩) ∨ (⟨1, 6⟩ ≠ ⟨0, 2⟩ ∧ ⟨1, 6⟩ ≠ ⟨1, 4⟩)) → {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩})) | |
23 | 7, 21, 22 | mp2 9 | . 2 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩} |
24 | zlmodzxzldep.a | . . 3 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩} | |
25 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩} | |
26 | 24, 25 | neeq12i 2999 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ {⟨0, 3⟩, ⟨1, 6⟩} ≠ {⟨0, 2⟩, ⟨1, 4⟩}) |
27 | 23, 26 | mpbir 230 | 1 ⊢ 𝐴 ≠ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 844 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 Vcvv 3466 {cpr 4622 ⟨cop 4626 (class class class)co 7401 0cc0 11106 1c1 11107 2c2 12264 3c3 12265 4c4 12266 6c6 12268 ℤringczring 21301 freeLMod cfrlm 21609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-2 12272 df-3 12273 |
This theorem is referenced by: zlmodzxzldeplem1 47369 zlmodzxzldeplem3 47371 zlmodzxzldeplem4 47372 ldepsnlinc 47377 |
Copyright terms: Public domain | W3C validator |