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Mirrors > Home > ILE Home > Th. List > rngsubdir | GIF version |
Description: Ring multiplication distributes over subtraction. (subdir 8405 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) Generalization of ringsubdir 13553. (Revised by AV, 23-Feb-2025.) |
Ref | Expression |
---|---|
rngsubdi.b | ⊢ 𝐵 = (Base‘𝑅) |
rngsubdi.t | ⊢ · = (.r‘𝑅) |
rngsubdi.m | ⊢ − = (-g‘𝑅) |
rngsubdi.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
rngsubdi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngsubdi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rngsubdi.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
rngsubdir | ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngsubdi.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
2 | rngsubdi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | rngsubdi.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2193 | . . . . 5 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
5 | rnggrp 13434 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
6 | 1, 5 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) |
7 | rngsubdi.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
8 | 3, 4, 6, 7 | grpinvcld 13121 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘𝑌) ∈ 𝐵) |
9 | rngsubdi.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
10 | eqid 2193 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
11 | rngsubdi.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
12 | 3, 10, 11 | rngdir 13437 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ ((invg‘𝑅)‘𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌)) · 𝑍) = ((𝑋 · 𝑍)(+g‘𝑅)(((invg‘𝑅)‘𝑌) · 𝑍))) |
13 | 1, 2, 8, 9, 12 | syl13anc 1251 | . . 3 ⊢ (𝜑 → ((𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌)) · 𝑍) = ((𝑋 · 𝑍)(+g‘𝑅)(((invg‘𝑅)‘𝑌) · 𝑍))) |
14 | 3, 11, 4, 1, 7, 9 | rngmneg1 13443 | . . . 4 ⊢ (𝜑 → (((invg‘𝑅)‘𝑌) · 𝑍) = ((invg‘𝑅)‘(𝑌 · 𝑍))) |
15 | 14 | oveq2d 5934 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑍)(+g‘𝑅)(((invg‘𝑅)‘𝑌) · 𝑍)) = ((𝑋 · 𝑍)(+g‘𝑅)((invg‘𝑅)‘(𝑌 · 𝑍)))) |
16 | 13, 15 | eqtrd 2226 | . 2 ⊢ (𝜑 → ((𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌)) · 𝑍) = ((𝑋 · 𝑍)(+g‘𝑅)((invg‘𝑅)‘(𝑌 · 𝑍)))) |
17 | rngsubdi.m | . . . . 5 ⊢ − = (-g‘𝑅) | |
18 | 3, 10, 4, 17 | grpsubval 13118 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌))) |
19 | 2, 7, 18 | syl2anc 411 | . . 3 ⊢ (𝜑 → (𝑋 − 𝑌) = (𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌))) |
20 | 19 | oveq1d 5933 | . 2 ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋(+g‘𝑅)((invg‘𝑅)‘𝑌)) · 𝑍)) |
21 | 3, 11 | rngcl 13440 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
22 | 1, 2, 9, 21 | syl3anc 1249 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
23 | 3, 11 | rngcl 13440 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 · 𝑍) ∈ 𝐵) |
24 | 1, 7, 9, 23 | syl3anc 1249 | . . 3 ⊢ (𝜑 → (𝑌 · 𝑍) ∈ 𝐵) |
25 | 3, 10, 4, 17 | grpsubval 13118 | . . 3 ⊢ (((𝑋 · 𝑍) ∈ 𝐵 ∧ (𝑌 · 𝑍) ∈ 𝐵) → ((𝑋 · 𝑍) − (𝑌 · 𝑍)) = ((𝑋 · 𝑍)(+g‘𝑅)((invg‘𝑅)‘(𝑌 · 𝑍)))) |
26 | 22, 24, 25 | syl2anc 411 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑍) − (𝑌 · 𝑍)) = ((𝑋 · 𝑍)(+g‘𝑅)((invg‘𝑅)‘(𝑌 · 𝑍)))) |
27 | 16, 20, 26 | 3eqtr4d 2236 | 1 ⊢ (𝜑 → ((𝑋 − 𝑌) · 𝑍) = ((𝑋 · 𝑍) − (𝑌 · 𝑍))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 ‘cfv 5254 (class class class)co 5918 Basecbs 12618 +gcplusg 12695 .rcmulr 12696 Grpcgrp 13072 invgcminusg 13073 -gcsg 13074 Rngcrng 13428 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-addcom 7972 ax-addass 7974 ax-i2m1 7977 ax-0lt1 7978 ax-0id 7980 ax-rnegex 7981 ax-pre-ltirr 7984 ax-pre-ltadd 7988 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-ltxr 8059 df-inn 8983 df-2 9041 df-3 9042 df-ndx 12621 df-slot 12622 df-base 12624 df-sets 12625 df-plusg 12708 df-mulr 12709 df-0g 12869 df-mgm 12939 df-sgrp 12985 df-mnd 12998 df-grp 13075 df-minusg 13076 df-sbg 13077 df-abl 13357 df-mgp 13417 df-rng 13429 |
This theorem is referenced by: 2idlcpblrng 14019 |
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