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Mirrors > Home > MPE Home > Th. List > ringadd2 | Structured version Visualization version GIF version |
Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (Revised by AV, 24-Aug-2021.) (Proof shortened by AV, 1-Feb-2025.) |
Ref | Expression |
---|---|
ringadd2.b | ⊢ 𝐵 = (Base‘𝑅) |
ringadd2.p | ⊢ + = (+g‘𝑅) |
ringadd2.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
ringadd2 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringadd2.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2736 | . . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | 1, 2 | ringidcl 19985 | . . 3 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
4 | 3 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (1r‘𝑅) ∈ 𝐵) |
5 | simpr 485 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = (1r‘𝑅)) → 𝑥 = (1r‘𝑅)) | |
6 | 5, 5 | oveq12d 7372 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = (1r‘𝑅)) → (𝑥 + 𝑥) = ((1r‘𝑅) + (1r‘𝑅))) |
7 | 6 | oveq1d 7369 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = (1r‘𝑅)) → ((𝑥 + 𝑥) · 𝑋) = (((1r‘𝑅) + (1r‘𝑅)) · 𝑋)) |
8 | 7 | eqeq2d 2747 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 = (1r‘𝑅)) → ((𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋) ↔ (𝑋 + 𝑋) = (((1r‘𝑅) + (1r‘𝑅)) · 𝑋))) |
9 | ringadd2.p | . . 3 ⊢ + = (+g‘𝑅) | |
10 | ringadd2.t | . . 3 ⊢ · = (.r‘𝑅) | |
11 | 1, 9, 10, 2 | ringo2times 19992 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 + 𝑋) = (((1r‘𝑅) + (1r‘𝑅)) · 𝑋)) |
12 | 4, 8, 11 | rspcedvd 3582 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∃𝑥 ∈ 𝐵 (𝑋 + 𝑋) = ((𝑥 + 𝑥) · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3072 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 +gcplusg 17130 .rcmulr 17131 1rcur 19909 Ringcrg 19960 |
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 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-plusg 17143 df-0g 17320 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-mgp 19893 df-ur 19910 df-ring 19962 |
This theorem is referenced by: (None) |
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