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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 48155 | . . . 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 3002 | . . . 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 3687 | . . . . . 6 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℤ) | |
| 9 | 8 | zcnd 12723 | . . . . 5 ⊢ (𝑏 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} → 𝑏 ∈ ℂ) | 
| 10 | 9, 1 | eleq2s 2859 | . . . 4 ⊢ (𝑏 ∈ 𝐸 → 𝑏 ∈ ℂ) | 
| 11 | 1 | 1neven 48154 | . . . . . . . 8 ⊢ 1 ∉ 𝐸 | 
| 12 | elnelne2 3058 | . . . . . . . 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 12344 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ∈ ℂ) | |
| 17 | 2ne0 12370 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
| 18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → 2 ≠ 0) | 
| 19 | 15, 16, 18 | divcan4d 12049 | . . . . . 6 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) = 𝑏) | 
| 20 | 2cnne0 12476 | . . . . . . 7 ⊢ (2 ∈ ℂ ∧ 2 ≠ 0) | |
| 21 | divid 11953 | . . . . . . 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 3018 | . . . . 5 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → ((𝑏 · 2) / 2) ≠ (2 / 2)) | 
| 24 | 15, 16 | mulcld 11281 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (𝑏 · 2) ∈ ℂ) | 
| 25 | 20 | a1i 11 | . . . . . . . 8 ⊢ ((𝑏 ∈ 𝐸 ∧ 𝑏 ∈ ℂ) → (2 ∈ ℂ ∧ 2 ≠ 0)) | 
| 26 | div11 11950 | . . . . . . . 8 ⊢ (((𝑏 · 2) ∈ ℂ ∧ 2 ∈ ℂ ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) → (((𝑏 · 2) / 2) = (2 / 2) ↔ (𝑏 · 2) = 2)) | |
| 27 | 24, 16, 25, 26 | syl3anc 1373 | . . . . . . 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 2961 | . . . . 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 3063 | 1 ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∉ wnel 3046 ∀wral 3061 ∃wrex 3070 {crab 3436 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 2c2 12321 ℤcz 12613 ↾s cress 17274 mulGrpcmgp 20137 ℂfldccnfld 21364 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 | 
| This theorem is referenced by: 2zrngnring 48174 | 
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