![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rngmneg1 | Structured version Visualization version GIF version |
Description: Negation of a product in a non-unital ring (mulneg1 11697 analog). In contrast to ringmneg1 20318, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.) |
Ref | Expression |
---|---|
rngneglmul.b | ⊢ 𝐵 = (Base‘𝑅) |
rngneglmul.t | ⊢ · = (.r‘𝑅) |
rngneglmul.n | ⊢ 𝑁 = (invg‘𝑅) |
rngneglmul.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
rngneglmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
rngneglmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
rngmneg1 | ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngneglmul.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
2 | eqid 2735 | . . . . . 6 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
3 | eqid 2735 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | rngneglmul.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
5 | rngneglmul.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
6 | rnggrp 20176 | . . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
8 | rngneglmul.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | 1, 2, 3, 4, 7, 8 | grprinvd 19026 | . . . . 5 ⊢ (𝜑 → (𝑋(+g‘𝑅)(𝑁‘𝑋)) = (0g‘𝑅)) |
10 | 9 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌) = ((0g‘𝑅) · 𝑌)) |
11 | rngneglmul.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
12 | rngneglmul.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
13 | 1, 12, 3 | rnglz 20183 | . . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑌 ∈ 𝐵) → ((0g‘𝑅) · 𝑌) = (0g‘𝑅)) |
14 | 5, 11, 13 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((0g‘𝑅) · 𝑌) = (0g‘𝑅)) |
15 | 10, 14 | eqtrd 2775 | . . 3 ⊢ (𝜑 → ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌) = (0g‘𝑅)) |
16 | 1, 12 | rngcl 20182 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
17 | 5, 8, 11, 16 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
18 | 1, 4, 7, 8 | grpinvcld 19019 | . . . . . 6 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝐵) |
19 | 1, 12 | rngcl 20182 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑁‘𝑋) · 𝑌) ∈ 𝐵) |
20 | 5, 18, 11, 19 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) ∈ 𝐵) |
21 | 1, 2, 3, 4 | grpinvid1 19022 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 𝑌) ∈ 𝐵 ∧ ((𝑁‘𝑋) · 𝑌) ∈ 𝐵) → ((𝑁‘(𝑋 · 𝑌)) = ((𝑁‘𝑋) · 𝑌) ↔ ((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌)) = (0g‘𝑅))) |
22 | 7, 17, 20, 21 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → ((𝑁‘(𝑋 · 𝑌)) = ((𝑁‘𝑋) · 𝑌) ↔ ((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌)) = (0g‘𝑅))) |
23 | 1, 2, 12 | rngdir 20179 | . . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌) = ((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌))) |
24 | 23 | eqcomd 2741 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ (𝑋 ∈ 𝐵 ∧ (𝑁‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌)) = ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌)) |
25 | 5, 8, 18, 11, 24 | syl13anc 1371 | . . . . 5 ⊢ (𝜑 → ((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌)) = ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌)) |
26 | 25 | eqeq1d 2737 | . . . 4 ⊢ (𝜑 → (((𝑋 · 𝑌)(+g‘𝑅)((𝑁‘𝑋) · 𝑌)) = (0g‘𝑅) ↔ ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌) = (0g‘𝑅))) |
27 | 22, 26 | bitrd 279 | . . 3 ⊢ (𝜑 → ((𝑁‘(𝑋 · 𝑌)) = ((𝑁‘𝑋) · 𝑌) ↔ ((𝑋(+g‘𝑅)(𝑁‘𝑋)) · 𝑌) = (0g‘𝑅))) |
28 | 15, 27 | mpbird 257 | . 2 ⊢ (𝜑 → (𝑁‘(𝑋 · 𝑌)) = ((𝑁‘𝑋) · 𝑌)) |
29 | 28 | eqcomd 2741 | 1 ⊢ (𝜑 → ((𝑁‘𝑋) · 𝑌) = (𝑁‘(𝑋 · 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 0gc0g 17486 Grpcgrp 18964 invgcminusg 18965 Rngcrng 20170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-abl 19816 df-mgp 20153 df-rng 20171 |
This theorem is referenced by: rngm2neg 20187 rngsubdir 20190 cntzsubrng 20584 |
Copyright terms: Public domain | W3C validator |