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Mirrors > Home > MPE Home > Th. List > ringmneg2 | Structured version Visualization version GIF version |
Description: Negation of a product in a ring. (mulneg2 11688 analog.) Compared with rngmneg2 20137, the proof is shorter making use of the existence of a ring unity. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringneglmul.b | ⊢ 𝐵 = (Base‘𝑅) |
ringneglmul.t | ⊢ · = (.r‘𝑅) |
ringneglmul.n | ⊢ 𝑁 = (invg‘𝑅) |
ringneglmul.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringneglmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringneglmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
ringmneg2 | ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringneglmul.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringneglmul.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringneglmul.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringgrp 20207 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringneglmul.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
7 | eqid 2725 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 6, 7 | ringidcl 20231 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
9 | 1, 8 | syl 17 | . . . 4 ⊢ (𝜑 → (1r‘𝑅) ∈ 𝐵) |
10 | ringneglmul.n | . . . . 5 ⊢ 𝑁 = (invg‘𝑅) | |
11 | 6, 10 | grpinvcl 18968 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (1r‘𝑅) ∈ 𝐵) → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
12 | 5, 9, 11 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑁‘(1r‘𝑅)) ∈ 𝐵) |
13 | ringneglmul.t | . . . 4 ⊢ · = (.r‘𝑅) | |
14 | 6, 13 | ringass 20222 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑁‘(1r‘𝑅)) ∈ 𝐵)) → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
15 | 1, 2, 3, 12, 14 | syl13anc 1369 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅))))) |
16 | 6, 13 | ringcl 20219 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
17 | 1, 2, 3, 16 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
18 | 6, 13, 7, 10, 1, 17 | ringnegr 20268 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) · (𝑁‘(1r‘𝑅))) = (𝑁‘(𝑋 · 𝑌))) |
19 | 6, 13, 7, 10, 1, 3 | ringnegr 20268 | . . 3 ⊢ (𝜑 → (𝑌 · (𝑁‘(1r‘𝑅))) = (𝑁‘𝑌)) |
20 | 19 | oveq2d 7435 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 · (𝑁‘(1r‘𝑅)))) = (𝑋 · (𝑁‘𝑌))) |
21 | 15, 18, 20 | 3eqtr3rd 2774 | 1 ⊢ (𝜑 → (𝑋 · (𝑁‘𝑌)) = (𝑁‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 .rcmulr 17253 Grpcgrp 18914 invgcminusg 18915 1rcur 20150 Ringcrg 20202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-plusg 17265 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 df-minusg 18918 df-cmn 19766 df-abl 19767 df-mgp 20104 df-rng 20122 df-ur 20151 df-ring 20204 |
This theorem is referenced by: cntzsubr 20574 abvneg 20743 erler 33076 lflnegcl 38697 |
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