Theorem 2zrngnmlid 40053

Description: R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)

Hypotheses

Ref	Expression
2zrng.e	|- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }
2zrngbas.r	|- R = (ℂfld ↾s E )
2zrngmmgm.1	|- M = (mulGrp ` R )

Assertion

Ref	Expression
2zrngnmlid	|- A. b e. E E. a e. E ( b x. a ) =/= a

Distinct variable groups:   x, z   E, a, b   R, a, b, x, z   x, E, z   M, a, b

Allowed substitution hints:   M( x, z)

Proof of Theorem 2zrngnmlid

Step	Hyp	Ref	Expression
1		2zrng.e	. . . . 5 |- E = { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) }
2	1	2even 40037	. . . 4 |- 2 e. E
3	2	a1i 11	. . 3 |- ( b e. E -> 2 e. E )
4		oveq2 6303	. . . . 5 |- ( a = 2 -> ( b x. a ) = ( b x. 2 ) )
5		id 22	. . . . 5 |- ( a = 2 -> a = 2 )
6	4, 5	neeq12d 2687	. . . 4 |- ( a = 2 -> ( ( b x. a ) =/= a <-> ( b x. 2 ) =/= 2 ) )
7	6	adantl 468	. . 3 |- ( ( b e. E /\ a = 2 ) -> ( ( b x. a ) =/= a <-> ( b x. 2 ) =/= 2 ) )
8		elrabi 3195	. . . . . 6 |- ( b e. { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) } -> b e. ZZ )
9	8	zcnd 11048	. . . . 5 |- ( b e. { z e. ZZ | E. x e. ZZ z = ( 2 x. x ) } -> b e. CC )
10	9, 1	eleq2s 2549	. . . 4 |- ( b e. E -> b e. CC )
11	1	1neven 40036	. . . . . . . 8 |- 1 e/ E
12		elnelne2 2737	. . . . . . . 8 |- ( ( b e. E /\ 1 e/ E ) -> b =/= 1 )
13	11, 12	mpan2 678	. . . . . . 7 |- ( b e. E -> b =/= 1 )
14	13	adantr 467	. . . . . 6 |- ( ( b e. E /\ b e. CC ) -> b =/= 1 )
15		simpr 463	. . . . . . 7 |- ( ( b e. E /\ b e. CC ) -> b e. CC )
16		2cnd 10689	. . . . . . 7 |- ( ( b e. E /\ b e. CC ) -> 2 e. CC )
17		2ne0 10709	. . . . . . . 8 |- 2 =/= 0
18	17	a1i 11	. . . . . . 7 |- ( ( b e. E /\ b e. CC ) -> 2 =/= 0 )
19	15, 16, 18	divcan4d 10396	. . . . . 6 |- ( ( b e. E /\ b e. CC ) -> ( ( b x. 2 ) / 2 ) = b )
20		2cnne0 10831	. . . . . . 7 |- ( 2 e. CC /\ 2 =/= 0 )
21		divid 10304	. . . . . . 7 |- ( ( 2 e. CC /\ 2 =/= 0 ) -> ( 2 / 2 ) = 1 )
22	20, 21	mp1i 13	. . . . . 6 |- ( ( b e. E /\ b e. CC ) -> ( 2 / 2 ) = 1 )
23	14, 19, 22	3netr4d 2703	. . . . 5 |- ( ( b e. E /\ b e. CC ) -> ( ( b x. 2 ) / 2 ) =/= ( 2 / 2 ) )
24	15, 16	mulcld 9668	. . . . . . . 8 |- ( ( b e. E /\ b e. CC ) -> ( b x. 2 ) e. CC )
25	20	a1i 11	. . . . . . . 8 |- ( ( b e. E /\ b e. CC ) -> ( 2 e. CC /\ 2 =/= 0 ) )
26		div11 10303	. . . . . . . 8 |- ( ( ( b x. 2 ) e. CC /\ 2 e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( ( b x. 2 ) / 2 ) = ( 2 / 2 ) <-> ( b x. 2 ) = 2 ) )
27	24, 16, 25, 26	syl3anc 1269	. . . . . . 7 |- ( ( b e. E /\ b e. CC ) -> ( ( ( b x. 2 ) / 2 ) = ( 2 / 2 ) <-> ( b x. 2 ) = 2 ) )
28	27	biimprd 227	. . . . . 6 |- ( ( b e. E /\ b e. CC ) -> ( ( b x. 2 ) = 2 -> ( ( b x. 2 ) / 2 ) = ( 2 / 2 ) ) )
29	28	necon3d 2647	. . . . 5 |- ( ( b e. E /\ b e. CC ) -> ( ( ( b x. 2 ) / 2 ) =/= ( 2 / 2 ) -> ( b x. 2 ) =/= 2 ) )
30	23, 29	mpd 15	. . . 4 |- ( ( b e. E /\ b e. CC ) -> ( b x. 2 ) =/= 2 )
31	10, 30	mpdan 675	. . 3 |- ( b e. E -> ( b x. 2 ) =/= 2 )
32	3, 7, 31	rspcedvd 3157	. 2 |- ( b e. E -> E. a e. E ( b x. a ) =/= a )
33	32	rgen 2749	1 |- A. b e. E E. a e. E ( b x. a ) =/= a

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   =/= wne 2624   e/ wnel 2625   A.wral 2739   E.wrex 2740   {crab 2743   ` cfv 5585  (class class class)co 6295   CCcc 9542   0cc0 9544   1c1 9545   x. cmul 9549   / cdiv 10276   2c2 10666   ZZcz 10944   ↾_s cress 15134  mulGrpcmgp 17735  ℂ_fldccnfld 18982

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945

This theorem is referenced by:  2zrngnring  40056