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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmodzxzldeplem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for zlmodzxzldep 46671. (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 |
---|---|
zlmodzxzldeplem4 | ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmodzxzldep.a | . . 3 ⊢ 𝐴 = {⟨0, 3⟩, ⟨1, 6⟩} | |
2 | prex 5390 | . . 3 ⊢ {⟨0, 3⟩, ⟨1, 6⟩} ∈ V | |
3 | 1, 2 | eqeltri 2830 | . 2 ⊢ 𝐴 ∈ V |
4 | zlmodzxzldep.b | . . 3 ⊢ 𝐵 = {⟨0, 2⟩, ⟨1, 4⟩} | |
5 | prex 5390 | . . 3 ⊢ {⟨0, 2⟩, ⟨1, 4⟩} ∈ V | |
6 | 4, 5 | eqeltri 2830 | . 2 ⊢ 𝐵 ∈ V |
7 | 2ne0 12262 | . . . . 5 ⊢ 2 ≠ 0 | |
8 | zlmodzxzldeplem.f | . . . . . . . 8 ⊢ 𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩} | |
9 | 8 | fveq1i 6844 | . . . . . . 7 ⊢ (𝐹‘𝐴) = ({⟨𝐴, 2⟩, ⟨𝐵, -3⟩}‘𝐴) |
10 | zlmodzxzldep.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤring freeLMod {0, 1}) | |
11 | 10, 1, 4 | zlmodzxzldeplem 46665 | . . . . . . . 8 ⊢ 𝐴 ≠ 𝐵 |
12 | 2ex 12235 | . . . . . . . . 9 ⊢ 2 ∈ V | |
13 | 3, 12 | fvpr1 7140 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, 2⟩, ⟨𝐵, -3⟩}‘𝐴) = 2) |
14 | 11, 13 | mp1i 13 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({⟨𝐴, 2⟩, ⟨𝐵, -3⟩}‘𝐴) = 2) |
15 | 9, 14 | eqtrid 2785 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) = 2) |
16 | 15 | neeq1d 3000 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ↔ 2 ≠ 0)) |
17 | 7, 16 | mpbiri 258 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐹‘𝐴) ≠ 0) |
18 | 17 | orcd 872 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0)) |
19 | fveq2 6843 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
20 | 19 | neeq1d 3000 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐴) ≠ 0)) |
21 | fveq2 6843 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵)) | |
22 | 21 | neeq1d 3000 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝑦) ≠ 0 ↔ (𝐹‘𝐵) ≠ 0)) |
23 | 20, 22 | rexprg 4658 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 ↔ ((𝐹‘𝐴) ≠ 0 ∨ (𝐹‘𝐵) ≠ 0))) |
24 | 18, 23 | mpbird 257 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0) |
25 | 3, 6, 24 | mp2an 691 | 1 ⊢ ∃𝑦 ∈ {𝐴, 𝐵} (𝐹‘𝑦) ≠ 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 Vcvv 3444 {cpr 4589 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 0cc0 11056 1c1 11057 -cneg 11391 2c2 12213 3c3 12214 4c4 12215 6c6 12217 ℤringczring 20885 freeLMod cfrlm 21168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-2 12221 df-3 12222 |
This theorem is referenced by: zlmodzxzldep 46671 |
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