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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for zlmodzxzldep 48350. (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 12620 | . 2 ⊢ ℤ ∈ V | |
2 | prex 5443 | . 2 ⊢ {𝐴, 𝐵} ∈ V | |
3 | zlmodzxzldep.a | . . . . . . . 8 ⊢ 𝐴 = {〈0, 3〉, 〈1, 6〉} | |
4 | prex 5443 | . . . . . . . 8 ⊢ {〈0, 3〉, 〈1, 6〉} ∈ V | |
5 | 3, 4 | eqeltri 2835 | . . . . . . 7 ⊢ 𝐴 ∈ V |
6 | zlmodzxzldep.b | . . . . . . . 8 ⊢ 𝐵 = {〈0, 2〉, 〈1, 4〉} | |
7 | prex 5443 | . . . . . . . 8 ⊢ {〈0, 2〉, 〈1, 4〉} ∈ V | |
8 | 6, 7 | eqeltri 2835 | . . . . . . 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 12647 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
12 | 3nn0 12542 | . . . . . . . 8 ⊢ 3 ∈ ℕ0 | |
13 | 12 | nn0negzi 12654 | . . . . . . 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 48344 | . . . . . 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 6728 | . . . . . 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 1370 | . . . 4 ⊢ ((ℤ ∈ V ∧ {𝐴, 𝐵} ∈ V) → 𝐹:{𝐴, 𝐵}⟶{2, -3}) |
24 | prssi 4826 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ -3 ∈ ℤ) → {2, -3} ⊆ ℤ) | |
25 | 11, 13, 24 | mp2an 692 | . . . 4 ⊢ {2, -3} ⊆ ℤ |
26 | fss 6753 | . . . 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 8878 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ⊆ wss 3963 {cpr 4633 〈cop 4637 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 0cc0 11153 1c1 11154 -cneg 11491 2c2 12319 3c3 12320 4c4 12321 6c6 12323 ℤcz 12611 ℤringczring 21475 freeLMod cfrlm 21784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 |
This theorem is referenced by: zlmodzxzldeplem2 48347 zlmodzxzldeplem3 48348 zlmodzxzldep 48350 |
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