Theorem ringinvnz1ne0 14061

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 6025	. . . . 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 14056	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
7	2, 6	sylan 283	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 )
8		eqeq12 2244	. . . . . . . 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 6025	. . . . 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 14036	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋)
17	3, 4, 5	ringrz 14056	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 )
18	16, 17	eqeq12d 2246	. . . . . . . 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 2676	. 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 ))
26	25	necon3bid 2443	1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∃wrex 2511  ‘cfv 5326  (class class class)co 6017  Basecbs 13081  ._rcmulr 13160  0_gc0g 13338  1_rcur 13971  Ringcrg 14008

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-mgp 13933  df-ur 13972  df-ring 14010

This theorem is referenced by: (None)