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| Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for zlmodzxzldep 48421. (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 12622 | . 2 ⊢ ℤ ∈ V | |
| 2 | prex 5437 | . 2 ⊢ {𝐴, 𝐵} ∈ V | |
| 3 | zlmodzxzldep.a | . . . . . . . 8 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
| 4 | prex 5437 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
| 5 | 3, 4 | eqeltri 2837 | . . . . . . 7 ⊢ 𝐴 ∈ V |
| 6 | zlmodzxzldep.b | . . . . . . . 8 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
| 7 | prex 5437 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
| 8 | 6, 7 | eqeltri 2837 | . . . . . . 7 ⊢ 𝐵 ∈ V |
| 9 | 5, 8 | pm3.2i 470 | . . . . . 6 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
| 10 | 9 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
| 11 | 2z 12649 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
| 12 | 3nn0 12544 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
| 13 | 12 | nn0negzi 12656 | . . . . . . 7 ⊢ -3 ∈ ℤ |
| 14 | 11, 13 | pm3.2i 470 | . . . . . 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 48415 | . . . . . 6 ⊢ 𝐴 ≠ 𝐵 |
| 18 | 17 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐴 ≠ 𝐵) |
| 19 | fprg 7175 | . . . . . 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 6727 | . . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}⟶{2, -3} ↔ {〈𝐴, 2〉, 〈𝐵, -3〉}:{𝐴, 𝐵}⟶{2, -3}) |
| 22 | 19, 21 | sylibr 234 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (2 ∈ ℤ ∧ -3 ∈ ℤ) ∧ 𝐴 ≠ 𝐵) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
| 23 | 10, 15, 18, 22 | syl3anc 1373 | . . . 4 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
| 24 | prssi 4821 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ -3 ∈ ℤ) → {2, -3} ⊆ ℤ) | |
| 25 | 11, 13, 24 | mp2an 692 | . . . 4 ⊢ {2, -3} ⊆ ℤ |
| 26 | fss 6752 | . . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶{2, -3} ∧ {2, -3} ⊆ ℤ) → 𝐹:{𝐴, 𝐵}⟶ℤ) | |
| 27 | 23, 25, 26 | sylancl 586 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶ℤ) |
| 28 | elmapg 8879 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐹 ∈ (ℤ ↑m {𝐴, 𝐵}) ↔ 𝐹:{𝐴, 𝐵}⟶ℤ)) | |
| 29 | 27, 28 | mpbird 257 | . 2 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵})) |
| 30 | 1, 2, 29 | mp2an 692 | 1 ⊢ 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ⊆ wss 3951 {cpr 4628 〈cop 4632 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 1c1 11156 -cneg 11493 2c2 12321 3c3 12322 4c4 12323 6c6 12325 ℤcz 12613 ℤringczring 21457 freeLMod cfrlm 21766 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 |
| This theorem is referenced by: zlmodzxzldeplem2 48418 zlmodzxzldeplem3 48419 zlmodzxzldep 48421 |
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