Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 42293. (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 |
---|---|
zlmodzxzldeplem1 | ⊢ 𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 11386 | . 2 ⊢ ℤ ∈ V | |
2 | prex 4909 | . 2 ⊢ {𝐴, 𝐵} ∈ V | |
3 | zlmodzxzldep.a | . . . . . . . 8 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
4 | prex 4909 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
5 | 3, 4 | eqeltri 2697 | . . . . . . 7 ⊢ 𝐴 ∈ V |
6 | zlmodzxzldep.b | . . . . . . . 8 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
7 | prex 4909 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
8 | 6, 7 | eqeltri 2697 | . . . . . . 7 ⊢ 𝐵 ∈ V |
9 | 5, 8 | pm3.2i 471 | . . . . . 6 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 2z 11409 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3nn0 11310 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
13 | 12 | nn0negzi 11416 | . . . . . . 7 ⊢ -3 ∈ ℤ |
14 | 11, 13 | pm3.2i 471 | . . . . . 6 ⊢ (2 ∈ ℤ ∧ -3 ∈ ℤ) |
15 | 14 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (2 ∈ ℤ ∧ -3 ∈ ℤ)) |
16 | zlmodzxzldep.z | . . . . . . 7 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
17 | 16, 3, 6 | zlmodzxzldeplem 42287 | . . . . . 6 ⊢ 𝐴 ≠ 𝐵 |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐴 ≠ 𝐵) |
19 | fprg 6422 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (2 ∈ ℤ ∧ -3 ∈ ℤ) ∧ 𝐴 ≠ 𝐵) → {〈𝐴, 2〉, 〈𝐵, -3〉}:{𝐴, 𝐵}⟶{2, -3}) | |
20 | zlmodzxzldeplem.f | . . . . . . 7 ⊢ 𝐹 = {〈𝐴, 2〉, 〈𝐵, -3〉} | |
21 | 20 | feq1i 6036 | . . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}⟶{2, -3} ↔ {〈𝐴, 2〉, 〈𝐵, -3〉}:{𝐴, 𝐵}⟶{2, -3}) |
22 | 19, 21 | sylibr 224 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (2 ∈ ℤ ∧ -3 ∈ ℤ) ∧ 𝐴 ≠ 𝐵) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
23 | 10, 15, 18, 22 | syl3anc 1326 | . . . 4 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
24 | prssi 4353 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ -3 ∈ ℤ) → {2, -3} ⊆ ℤ) | |
25 | 11, 13, 24 | mp2an 708 | . . . 4 ⊢ {2, -3} ⊆ ℤ |
26 | fss 6056 | . . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶{2, -3} ∧ {2, -3} ⊆ ℤ) → 𝐹:{𝐴, 𝐵}⟶ℤ) | |
27 | 23, 25, 26 | sylancl 694 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶ℤ) |
28 | elmapg 7870 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵}) ↔ 𝐹:{𝐴, 𝐵}⟶ℤ)) | |
29 | 27, 28 | mpbird 247 | . 2 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵})) |
30 | 1, 2, 29 | mp2an 708 | 1 ⊢ 𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ⊆ wss 3574 {cpr 4179 〈cop 4183 ⟶wf 5884 (class class class)co 6650 ↑𝑚 cmap 7857 0cc0 9936 1c1 9937 -cneg 10267 2c2 11070 3c3 11071 4c4 11072 6c6 11074 ℤcz 11377 ℤringzring 19818 freeLMod cfrlm 20090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 |
This theorem is referenced by: zlmodzxzldeplem2 42290 zlmodzxzldeplem3 42291 zlmodzxzldep 42293 |
Copyright terms: Public domain | W3C validator |