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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 44579. (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 11991 | . 2 ⊢ ℤ ∈ V | |
2 | prex 5333 | . 2 ⊢ {𝐴, 𝐵} ∈ V | |
3 | zlmodzxzldep.a | . . . . . . . 8 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
4 | prex 5333 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
5 | 3, 4 | eqeltri 2909 | . . . . . . 7 ⊢ 𝐴 ∈ V |
6 | zlmodzxzldep.b | . . . . . . . 8 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
7 | prex 5333 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
8 | 6, 7 | eqeltri 2909 | . . . . . . 7 ⊢ 𝐵 ∈ V |
9 | 5, 8 | pm3.2i 473 | . . . . . 6 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V) |
10 | 9 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
11 | 2z 12015 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3nn0 11916 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
13 | 12 | nn0negzi 12022 | . . . . . . 7 ⊢ -3 ∈ ℤ |
14 | 11, 13 | pm3.2i 473 | . . . . . 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 44573 | . . . . . 6 ⊢ 𝐴 ≠ 𝐵 |
18 | 17 | a1i 11 | . . . . 5 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐴 ≠ 𝐵) |
19 | fprg 6917 | . . . . . 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 6505 | . . . . . 6 ⊢ (𝐹:{𝐴, 𝐵}⟶{2, -3} ↔ {〈𝐴, 2〉, 〈𝐵, -3〉}:{𝐴, 𝐵}⟶{2, -3}) |
22 | 19, 21 | sylibr 236 | . . . . 5 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (2 ∈ ℤ ∧ -3 ∈ ℤ) ∧ 𝐴 ≠ 𝐵) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
23 | 10, 15, 18, 22 | syl3anc 1367 | . . . 4 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
24 | prssi 4754 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ -3 ∈ ℤ) → {2, -3} ⊆ ℤ) | |
25 | 11, 13, 24 | mp2an 690 | . . . 4 ⊢ {2, -3} ⊆ ℤ |
26 | fss 6527 | . . . 4 ⊢ ((𝐹:{𝐴, 𝐵}⟶{2, -3} ∧ {2, -3} ⊆ ℤ) → 𝐹:{𝐴, 𝐵}⟶ℤ) | |
27 | 23, 25, 26 | sylancl 588 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶ℤ) |
28 | elmapg 8419 | . . 3 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → (𝐹 ∈ (ℤ ↑m {𝐴, 𝐵}) ↔ 𝐹:{𝐴, 𝐵}⟶ℤ)) | |
29 | 27, 28 | mpbird 259 | . 2 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵})) |
30 | 1, 2, 29 | mp2an 690 | 1 ⊢ 𝐹 ∈ (ℤ ↑m {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 {cpr 4569 〈cop 4573 ⟶wf 6351 (class class class)co 7156 ↑m cmap 8406 0cc0 10537 1c1 10538 -cneg 10871 2c2 11693 3c3 11694 4c4 11695 6c6 11697 ℤcz 11982 ℤringzring 20617 freeLMod cfrlm 20890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 |
This theorem is referenced by: zlmodzxzldeplem2 44576 zlmodzxzldeplem3 44577 zlmodzxzldep 44579 |
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