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Mirrors > Home > MPE Home > Th. List > ringinvnz1ne0 | Structured version Visualization version GIF version |
Description: In a unitary ring, a left invertible element is different from zero iff 1 ≠ 0. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.) |
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 ) |
Ref | Expression |
---|---|
ringinvnz1ne0 | ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringinvnzdiv.a | . . 3 ⊢ (𝜑 → ∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 ) | |
2 | oveq2 6557 | . . . . . . 7 ⊢ (𝑋 = 0 → (𝑎 · 𝑋) = (𝑎 · 0 )) | |
3 | ringinvnzdiv.r | . . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
4 | ringinvnzdiv.b | . . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅) | |
5 | ringinvnzdiv.t | . . . . . . . . . 10 ⊢ · = (.r‘𝑅) | |
6 | ringinvnzdiv.z | . . . . . . . . . 10 ⊢ 0 = (0g‘𝑅) | |
7 | 4, 5, 6 | ringrz 18411 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
8 | 3, 7 | sylan 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 · 0 ) = 0 ) |
9 | eqeq12 2623 | . . . . . . . . . 10 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) ↔ 1 = 0 )) | |
10 | 9 | biimpd 218 | . . . . . . . . 9 ⊢ (((𝑎 · 𝑋) = 1 ∧ (𝑎 · 0 ) = 0 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 )) |
11 | 10 | ex 449 | . . . . . . . 8 ⊢ ((𝑎 · 𝑋) = 1 → ((𝑎 · 0 ) = 0 → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 ))) |
12 | 8, 11 | mpan9 485 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑎 · 𝑋) = (𝑎 · 0 ) → 1 = 0 )) |
13 | 2, 12 | syl5 33 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 → 1 = 0 )) |
14 | oveq2 6557 | . . . . . . 7 ⊢ ( 1 = 0 → (𝑋 · 1 ) = (𝑋 · 0 )) | |
15 | ringinvnzdiv.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
16 | ringinvnzdiv.u | . . . . . . . . . . . 12 ⊢ 1 = (1r‘𝑅) | |
17 | 4, 5, 16 | ringridm 18395 | . . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 1 ) = 𝑋) |
18 | 4, 5, 6 | ringrz 18411 | . . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 · 0 ) = 0 ) |
19 | 17, 18 | eqeq12d 2625 | . . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) ↔ 𝑋 = 0 )) |
20 | 19 | biimpd 218 | . . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
21 | 3, 15, 20 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
22 | 21 | ad2antrr 758 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ((𝑋 · 1 ) = (𝑋 · 0 ) → 𝑋 = 0 )) |
23 | 14, 22 | syl5 33 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → ( 1 = 0 → 𝑋 = 0 )) |
24 | 13, 23 | impbid 201 | . . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐵) ∧ (𝑎 · 𝑋) = 1 ) → (𝑋 = 0 ↔ 1 = 0 )) |
25 | 24 | ex 449 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑎 · 𝑋) = 1 → (𝑋 = 0 ↔ 1 = 0 ))) |
26 | 25 | rexlimdva 3013 | . . 3 ⊢ (𝜑 → (∃𝑎 ∈ 𝐵 (𝑎 · 𝑋) = 1 → (𝑋 = 0 ↔ 1 = 0 ))) |
27 | 1, 26 | mpd 15 | . 2 ⊢ (𝜑 → (𝑋 = 0 ↔ 1 = 0 )) |
28 | 27 | necon3bid 2826 | 1 ⊢ (𝜑 → (𝑋 ≠ 0 ↔ 1 ≠ 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 0gc0g 15923 1rcur 18324 Ringcrg 18370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-mgp 18313 df-ur 18325 df-ring 18372 |
This theorem is referenced by: (None) |
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