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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2zrngnmlid2 | 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 |
---|---|
2zrngnmlid2 | ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2zrng.e | . . 3 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
2 | 2zrngbas.r | . . 3 ⊢ 𝑅 = (ℂfld ↾s 𝐸) | |
3 | 2zrngmmgm.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
4 | 1, 2, 3 | 2zrngnmrid 47981 | . 2 ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑎 · 𝑏) ≠ 𝑎 |
5 | eldifi 4154 | . . . . . . . . . 10 ⊢ (𝑎 ∈ (𝐸 ∖ {0}) → 𝑎 ∈ 𝐸) | |
6 | elrabi 3703 | . . . . . . . . . . . 12 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℤ) | |
7 | 6 | zcnd 12750 | . . . . . . . . . . 11 ⊢ (𝑎 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑎 ∈ ℂ) |
8 | 7, 1 | eleq2s 2862 | . . . . . . . . . 10 ⊢ (𝑎 ∈ 𝐸 → 𝑎 ∈ ℂ) |
9 | 5, 8 | syl 17 | . . . . . . . . 9 ⊢ (𝑎 ∈ (𝐸 ∖ {0}) → 𝑎 ∈ ℂ) |
10 | elrabi 3703 | . . . . . . . . . . 11 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
11 | 10 | zcnd 12750 | . . . . . . . . . 10 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) |
12 | 11, 1 | eleq2s 2862 | . . . . . . . . 9 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) |
13 | mulcom 11272 | . . . . . . . . 9 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 · 𝑏) = (𝑏 · 𝑎)) | |
14 | 9, 12, 13 | syl2an 595 | . . . . . . . 8 ⊢ ((𝑎 ∈ (𝐸 ∖ {0}) ∧ 𝑏 ∈ 𝐸) → (𝑎 · 𝑏) = (𝑏 · 𝑎)) |
15 | 14 | eqcomd 2746 | . . . . . . 7 ⊢ ((𝑎 ∈ (𝐸 ∖ {0}) ∧ 𝑏 ∈ 𝐸) → (𝑏 · 𝑎) = (𝑎 · 𝑏)) |
16 | 15 | eqeq1d 2742 | . . . . . 6 ⊢ ((𝑎 ∈ (𝐸 ∖ {0}) ∧ 𝑏 ∈ 𝐸) → ((𝑏 · 𝑎) = 𝑎 ↔ (𝑎 · 𝑏) = 𝑎)) |
17 | 16 | biimpd 229 | . . . . 5 ⊢ ((𝑎 ∈ (𝐸 ∖ {0}) ∧ 𝑏 ∈ 𝐸) → ((𝑏 · 𝑎) = 𝑎 → (𝑎 · 𝑏) = 𝑎)) |
18 | 17 | necon3d 2967 | . . . 4 ⊢ ((𝑎 ∈ (𝐸 ∖ {0}) ∧ 𝑏 ∈ 𝐸) → ((𝑎 · 𝑏) ≠ 𝑎 → (𝑏 · 𝑎) ≠ 𝑎)) |
19 | 18 | ralimdva 3173 | . . 3 ⊢ (𝑎 ∈ (𝐸 ∖ {0}) → (∀𝑏 ∈ 𝐸 (𝑎 · 𝑏) ≠ 𝑎 → ∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎)) |
20 | 19 | ralimia 3086 | . 2 ⊢ (∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑎 · 𝑏) ≠ 𝑎 → ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎) |
21 | 4, 20 | ax-mp 5 | 1 ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ∖ cdif 3973 {csn 4648 ‘cfv 6575 (class class class)co 7450 ℂcc 11184 0cc0 11186 · cmul 11191 2c2 12350 ℤcz 12641 ↾s cress 17289 mulGrpcmgp 20163 ℂfldccnfld 21389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-er 8765 df-en 9006 df-dom 9007 df-sdom 9008 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-n0 12556 df-z 12642 |
This theorem is referenced by: (None) |
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