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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngnmlid | Structured version Visualization version GIF version |
Description: R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
2zrngbas.r | ⊢ 𝑅 = (ℂfld ↾s 𝐸) |
2zrngmmgm.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
2zrngnmlid | ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . . . 5 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | 1 | 2even 42258 | . . . 4 ⊢ 2 ∈ 𝐸 |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑏 ∈ 𝐸 → 2 ∈ 𝐸) |
4 | oveq2 6698 | . . . . 5 ⊢ (𝑎 = 2 → (𝑏 · 𝑎) = (𝑏 · 2)) | |
5 | id 22 | . . . . 5 ⊢ (𝑎 = 2 → 𝑎 = 2) | |
6 | 4, 5 | neeq12d 2884 | . . . 4 ⊢ (𝑎 = 2 → ((𝑏 · 𝑎) ≠ 𝑎 ↔ (𝑏 · 2) ≠ 2)) |
7 | 6 | adantl 481 | . . 3 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑎 = 2) → ((𝑏 · 𝑎) ≠ 𝑎 ↔ (𝑏 · 2) ≠ 2)) |
8 | elrabi 3391 | . . . . . 6 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
9 | 8 | zcnd 11521 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) |
10 | 9, 1 | eleq2s 2748 | . . . 4 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) |
11 | 1 | 1neven 42257 | . . . . . . . 8 ⊢ 1 ∉ 𝐸 |
12 | elnelne2 2937 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 1 ∉ 𝐸) → 𝑏 ≠ 1) | |
13 | 11, 12 | mpan2 707 | . . . . . . 7 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ≠ 1) |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 𝑏 ≠ 1) |
15 | simpr 476 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ) | |
16 | 2cnd 11131 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ∈ ℂ) | |
17 | 2ne0 11151 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ≠ 0) |
19 | 15, 16, 18 | divcan4d 10845 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) = 𝑏) |
20 | 2cnne0 11280 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
21 | divid 10752 | . . . . . . 7 ⊢ ((2 ∈ ℂ ∧ 2 ≠ 0) → (2 / 2) = 1) | |
22 | 20, 21 | mp1i 13 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (2 / 2) = 1) |
23 | 14, 19, 22 | 3netr4d 2900 | . . . . 5 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) ≠ (2 / 2)) |
24 | 15, 16 | mulcld 10098 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (𝑏 · 2) ∈ ℂ) |
25 | 20 | a1i 11 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
26 | div11 10751 | . . . . . . . 8 ⊢ (((𝑏 · 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((𝑏 · 2) / 2) = (2 / 2) ↔ (𝑏 · 2) = 2)) | |
27 | 24, 16, 25, 26 | syl3anc 1366 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (((𝑏 · 2) / 2) = (2 / 2) ↔ (𝑏 · 2) = 2)) |
28 | 27 | biimprd 238 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) = 2 → ((𝑏 · 2) / 2) = (2 / 2))) |
29 | 28 | necon3d 2844 | . . . . 5 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (((𝑏 · 2) / 2) ≠ (2 / 2) → (𝑏 · 2) ≠ 2)) |
30 | 23, 29 | mpd 15 | . . . 4 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (𝑏 · 2) ≠ 2) |
31 | 10, 30 | mpdan 703 | . . 3 ⊢ (𝑏 ∈ 𝐸 → (𝑏 · 2) ≠ 2) |
32 | 3, 7, 31 | rspcedvd 3348 | . 2 ⊢ (𝑏 ∈ 𝐸 → ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎) |
33 | 32 | rgen 2951 | 1 ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∉ wnel 2926 ∀wral 2941 ∃wrex 2942 {crab 2945 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 1c1 9975 · cmul 9979 / cdiv 10722 2c2 11108 ℤcz 11415 ↾s cress 15905 mulGrpcmgp 18535 ℂfldccnfld 19794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 |
This theorem is referenced by: 2zrngnring 42277 |
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