Theorem ringinvnz1ne0 14007

Description: In a unital ring, a left invertible element is different from zero iff 1 ≠ 0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)

Hypotheses

Ref	Expression
ringinvnzdiv.b	⊢ 𝐵 = (Base‘𝑅)
ringinvnzdiv.t	⊢ · = (.r‘𝑅)
ringinvnzdiv.u	⊢ 1 = (1r‘𝑅)
ringinvnzdiv.z	⊢ 0 = (0g‘𝑅)
ringinvnzdiv.r	⊢ (𝜑 → 𝑅 ∈ Ring)
ringinvnzdiv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ringinvnzdiv.a	⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 )

Assertion

Ref	Expression
ringinvnz1ne0	⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 ))

Distinct variable groups:   𝑋,𝑎   0 ,𝑎   1 ,𝑎   · ,𝑎   𝜑,𝑎

Allowed substitution hints:   𝐵(𝑎)   𝑅(𝑎)

Proof of Theorem ringinvnz1ne0

Step	Hyp	Ref	Expression
1		oveq2 6008	. . . . 5 ⊢ (𝑋 = 0 → (𝑎 · 𝑋) = (𝑎 · 0 ))
2		ringinvnzdiv.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
3		ringinvnzdiv.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
4		ringinvnzdiv.t	. . . . . . . 8 ⊢ · = (.r‘𝑅)
5		ringinvnzdiv.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑅)
6	3, 4, 5	ringrz 14002	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
7	2, 6	sylan 283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
8		eqeq12 2242	. . . . . . . 8 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) ↔ 1 = 0 ))
9	8	biimpd 144	. . . . . . 7 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 ))
10	9	ex 115	. . . . . 6 ⊢ ((𝑎 · 𝑋) = 1 → ((𝑎 · 0 ) = 0 → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 )))
11	7, 10	mpan9 281	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 ))
12	1, 11	syl5 32	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 → 1 = 0 ))
13		oveq2 6008	. . . . 5 ⊢ ( 1 = 0 → (𝑋 · 1 ) = (𝑋 · 0 ))
14		ringinvnzdiv.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15		ringinvnzdiv.u	. . . . . . . . . 10 ⊢ 1 = (1r‘𝑅)
16	3, 4, 15	ringridm 13982	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋)
17	3, 4, 5	ringrz 14002	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
18	16, 17	eqeq12d 2244	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) ↔ 𝑋 = 0 ))
19	18	biimpd 144	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 ))
20	2, 14, 19	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 ))
21	20	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 ))
22	13, 21	syl5 32	. . . 4 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ( 1 = 0 → 𝑋 = 0 ))
23	12, 22	impbid 129	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 ↔ 1 = 0 ))
24		ringinvnzdiv.a	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 )
25	23, 24	r19.29a 2674	. 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 ))
26	25	necon3bid 2441	1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∃wrex 2509  ‘cfv 5317  (class class class)co 6000  Basecbs 13027  ._rcmulr 13106  0_gc0g 13284  1_rcur 13917  Ringcrg 13954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-grp 13531  df-mgp 13879  df-ur 13918  df-ring 13956

This theorem is referenced by: (None)