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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 48083 | . . . 4 ⊢ 2 ∈ 𝐸 |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑏 ∈ 𝐸 → 2 ∈ 𝐸) |
4 | oveq2 7439 | . . . . 5 ⊢ (𝑎 = 2 → (𝑏 · 𝑎) = (𝑏 · 2)) | |
5 | id 22 | . . . . 5 ⊢ (𝑎 = 2 → 𝑎 = 2) | |
6 | 4, 5 | neeq12d 3000 | . . . 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 3690 | . . . . . 6 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
9 | 8 | zcnd 12721 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) |
10 | 9, 1 | eleq2s 2857 | . . . 4 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) |
11 | 1 | 1neven 48082 | . . . . . . . 8 ⊢ 1 ∉ 𝐸 |
12 | elnelne2 3056 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 1 ∉ 𝐸) → 𝑏 ≠ 1) | |
13 | 11, 12 | mpan2 691 | . . . . . . 7 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ≠ 1) |
14 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 𝑏 ≠ 1) |
15 | simpr 484 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 𝑏 ∈ ℂ) | |
16 | 2cnd 12342 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ∈ ℂ) | |
17 | 2ne0 12368 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ≠ 0) |
19 | 15, 16, 18 | divcan4d 12047 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) = 𝑏) |
20 | 2cnne0 12474 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
21 | divid 11951 | . . . . . . 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 3016 | . . . . 5 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) ≠ (2 / 2)) |
24 | 15, 16 | mulcld 11279 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (𝑏 · 2) ∈ ℂ) |
25 | 20 | a1i 11 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (2 ∈ ℂ ∧ 2 ≠ 0)) |
26 | div11 11948 | . . . . . . . 8 ⊢ (((𝑏 · 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((𝑏 · 2) / 2) = (2 / 2) ↔ (𝑏 · 2) = 2)) | |
27 | 24, 16, 25, 26 | syl3anc 1370 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (((𝑏 · 2) / 2) = (2 / 2) ↔ (𝑏 · 2) = 2)) |
28 | 27 | biimprd 248 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) = 2 → ((𝑏 · 2) / 2) = (2 / 2))) |
29 | 28 | necon3d 2959 | . . . . 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 687 | . . 3 ⊢ (𝑏 ∈ 𝐸 → (𝑏 · 2) ≠ 2) |
32 | 3, 7, 31 | rspcedvd 3624 | . 2 ⊢ (𝑏 ∈ 𝐸 → ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎) |
33 | 32 | rgen 3061 | 1 ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∉ wnel 3044 ∀wral 3059 ∃wrex 3068 {crab 3433 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 / cdiv 11918 2c2 12319 ℤcz 12611 ↾s cress 17274 mulGrpcmgp 20152 ℂfldccnfld 21382 |
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-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-rmo 3378 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-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-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 |
This theorem is referenced by: 2zrngnring 48102 |
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