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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 43956. (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 11801 | . 2 ⊢ ℤ ∈ V | |
2 | prex 5186 | . 2 ⊢ {𝐴, 𝐵} ∈ V | |
3 | zlmodzxzldep.a | . . . . . . . 8 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
4 | prex 5186 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
5 | 3, 4 | eqeltri 2857 | . . . . . . 7 ⊢ 𝐴 ∈ V |
6 | zlmodzxzldep.b | . . . . . . . 8 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
7 | prex 5186 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
8 | 6, 7 | eqeltri 2857 | . . . . . . 7 ⊢ 𝐵 ∈ V |
9 | 5, 8 | pm3.2i 463 | . . . . . 6 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 2z 11826 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3nn0 11726 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
13 | 12 | nn0negzi 11833 | . . . . . . 7 ⊢ -3 ∈ ℤ |
14 | 11, 13 | pm3.2i 463 | . . . . . 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 43950 | . . . . . 6 ⊢ 𝐴 ≠ 𝐵 |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐴 ≠ 𝐵) |
19 | fprg 6739 | . . . . . 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 6333 | . . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}⟶{2, -3} ↔ {〈𝐴, 2〉, 〈𝐵, -3〉}:{𝐴, 𝐵}⟶{2, -3}) |
22 | 19, 21 | sylibr 226 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (2 ∈ ℤ ∧ -3 ∈ ℤ) ∧ 𝐴 ≠ 𝐵) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
23 | 10, 15, 18, 22 | syl3anc 1352 | . . . 4 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
24 | prssi 4625 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ -3 ∈ ℤ) → {2, -3} ⊆ ℤ) | |
25 | 11, 13, 24 | mp2an 680 | . . . 4 ⊢ {2, -3} ⊆ ℤ |
26 | fss 6355 | . . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶{2, -3} ∧ {2, -3} ⊆ ℤ) → 𝐹:{𝐴, 𝐵}⟶ℤ) | |
27 | 23, 25, 26 | sylancl 578 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶ℤ) |
28 | elmapg 8218 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵}) ↔ 𝐹:{𝐴, 𝐵}⟶ℤ)) | |
29 | 27, 28 | mpbird 249 | . 2 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵})) |
30 | 1, 2, 29 | mp2an 680 | 1 ⊢ 𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∈ wcel 2051 ≠ wne 2962 Vcvv 3410 ⊆ wss 3824 {cpr 4438 〈cop 4442 ⟶wf 6182 (class class class)co 6975 ↑𝑚 cmap 8205 0cc0 10334 1c1 10335 -cneg 10670 2c2 11494 3c3 11495 4c4 11496 6c6 11498 ℤcz 11792 ℤringzring 20335 freeLMod cfrlm 20608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-map 8207 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-n0 11707 df-z 11793 |
This theorem is referenced by: zlmodzxzldeplem2 43953 zlmodzxzldeplem3 43954 zlmodzxzldep 43956 |
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