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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringinveu | Structured version Visualization version GIF version |
Description: If a ring unit element 𝑋 admits both a left inverse 𝑌 and a right inverse 𝑍, they are equal. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
isdrng4.b | ⊢ 𝐵 = (Base‘𝑅) |
isdrng4.0 | ⊢ 0 = (0g‘𝑅) |
isdrng4.1 | ⊢ 1 = (1r‘𝑅) |
isdrng4.x | ⊢ · = (.r‘𝑅) |
isdrng4.u | ⊢ 𝑈 = (Unit‘𝑅) |
isdrng4.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringinveu.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringinveu.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringinveu.3 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ringinveu.4 | ⊢ (𝜑 → (𝑌 · 𝑋) = 1 ) |
ringinveu.5 | ⊢ (𝜑 → (𝑋 · 𝑍) = 1 ) |
Ref | Expression |
---|---|
ringinveu | ⊢ (𝜑 → 𝑍 = 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringinveu.5 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) = 1 ) | |
2 | 1 | oveq2d 7410 | . 2 ⊢ (𝜑 → (𝑌 · (𝑋 · 𝑍)) = (𝑌 · 1 )) |
3 | ringinveu.4 | . . . 4 ⊢ (𝜑 → (𝑌 · 𝑋) = 1 ) | |
4 | 3 | oveq1d 7409 | . . 3 ⊢ (𝜑 → ((𝑌 · 𝑋) · 𝑍) = ( 1 · 𝑍)) |
5 | isdrng4.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | isdrng4.x | . . . 4 ⊢ · = (.r‘𝑅) | |
7 | isdrng4.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | ringinveu.2 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | ringinveu.1 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | ringinveu.3 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | 5, 6, 7, 8, 9, 10 | ringassd 20038 | . . 3 ⊢ (𝜑 → ((𝑌 · 𝑋) · 𝑍) = (𝑌 · (𝑋 · 𝑍))) |
12 | isdrng4.1 | . . . 4 ⊢ 1 = (1r‘𝑅) | |
13 | 5, 6, 12, 7, 10 | ringlidmd 20048 | . . 3 ⊢ (𝜑 → ( 1 · 𝑍) = 𝑍) |
14 | 4, 11, 13 | 3eqtr3d 2780 | . 2 ⊢ (𝜑 → (𝑌 · (𝑋 · 𝑍)) = 𝑍) |
15 | 5, 6, 12, 7, 8 | ringridmd 20049 | . 2 ⊢ (𝜑 → (𝑌 · 1 ) = 𝑌) |
16 | 2, 14, 15 | 3eqtr3d 2780 | 1 ⊢ (𝜑 → 𝑍 = 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 .rcmulr 17182 0gc0g 17369 1rcur 19965 Ringcrg 20016 Unitcui 20123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-er 8688 df-en 8925 df-dom 8926 df-sdom 8927 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-plusg 17194 df-0g 17371 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-mgp 19949 df-ur 19966 df-ring 20018 |
This theorem is referenced by: isdrng4 32321 drngidl 32466 qsdrngi 32519 |
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