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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unitmulrprm | Structured version Visualization version GIF version |
Description: A ring unit multiplied by a ring prime is a ring prime. (Contributed by Thierry Arnoux, 3-Jun-2025.) |
Ref | Expression |
---|---|
unitmulrprm.p | ⊢ 𝑃 = (RPrime‘𝑅) |
unitmulrprm.u | ⊢ 𝑈 = (Unit‘𝑅) |
unitmulrprm.t | ⊢ · = (.r‘𝑅) |
unitmulrprm.r | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
unitmulrprm.i | ⊢ (𝜑 → 𝐼 ∈ 𝑈) |
unitmulrprm.q | ⊢ (𝜑 → 𝑄 ∈ 𝑃) |
Ref | Expression |
---|---|
unitmulrprm | ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2735 | . 2 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | unitmulrprm.p | . 2 ⊢ 𝑃 = (RPrime‘𝑅) | |
3 | eqid 2735 | . 2 ⊢ (∥r‘𝑅) = (∥r‘𝑅) | |
4 | unitmulrprm.r | . 2 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
5 | unitmulrprm.q | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝑃) | |
6 | 1, 2, 4, 5 | rprmcl 33526 | . . 3 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝑅)) |
7 | oveq1 7438 | . . . . 5 ⊢ (𝑖 = 𝐼 → (𝑖 · 𝑄) = (𝐼 · 𝑄)) | |
8 | 7 | eqeq1d 2737 | . . . 4 ⊢ (𝑖 = 𝐼 → ((𝑖 · 𝑄) = (𝐼 · 𝑄) ↔ (𝐼 · 𝑄) = (𝐼 · 𝑄))) |
9 | unitmulrprm.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑈) | |
10 | unitmulrprm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
11 | 1, 10 | unitcl 20392 | . . . . 5 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ (Base‘𝑅)) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (Base‘𝑅)) |
13 | eqidd 2736 | . . . 4 ⊢ (𝜑 → (𝐼 · 𝑄) = (𝐼 · 𝑄)) | |
14 | 8, 12, 13 | rspcedvdw 3625 | . . 3 ⊢ (𝜑 → ∃𝑖 ∈ (Base‘𝑅)(𝑖 · 𝑄) = (𝐼 · 𝑄)) |
15 | unitmulrprm.t | . . . 4 ⊢ · = (.r‘𝑅) | |
16 | 1, 3, 15 | dvdsr 20379 | . . 3 ⊢ (𝑄(∥r‘𝑅)(𝐼 · 𝑄) ↔ (𝑄 ∈ (Base‘𝑅) ∧ ∃𝑖 ∈ (Base‘𝑅)(𝑖 · 𝑄) = (𝐼 · 𝑄))) |
17 | 6, 14, 16 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝑄(∥r‘𝑅)(𝐼 · 𝑄)) |
18 | 4 | idomringd 20745 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) |
19 | 1, 15, 18, 12, 6 | ringcld 20277 | . . 3 ⊢ (𝜑 → (𝐼 · 𝑄) ∈ (Base‘𝑅)) |
20 | oveq1 7438 | . . . . 5 ⊢ (𝑖 = ((invr‘𝑅)‘𝐼) → (𝑖 · (𝐼 · 𝑄)) = (((invr‘𝑅)‘𝐼) · (𝐼 · 𝑄))) | |
21 | 20 | eqeq1d 2737 | . . . 4 ⊢ (𝑖 = ((invr‘𝑅)‘𝐼) → ((𝑖 · (𝐼 · 𝑄)) = 𝑄 ↔ (((invr‘𝑅)‘𝐼) · (𝐼 · 𝑄)) = 𝑄)) |
22 | eqid 2735 | . . . . . 6 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
23 | 10, 22, 1 | ringinvcl 20409 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → ((invr‘𝑅)‘𝐼) ∈ (Base‘𝑅)) |
24 | 18, 9, 23 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((invr‘𝑅)‘𝐼) ∈ (Base‘𝑅)) |
25 | eqid 2735 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
26 | 10, 22, 15, 25 | unitlinv 20410 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑈) → (((invr‘𝑅)‘𝐼) · 𝐼) = (1r‘𝑅)) |
27 | 18, 9, 26 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (((invr‘𝑅)‘𝐼) · 𝐼) = (1r‘𝑅)) |
28 | 27 | oveq1d 7446 | . . . . 5 ⊢ (𝜑 → ((((invr‘𝑅)‘𝐼) · 𝐼) · 𝑄) = ((1r‘𝑅) · 𝑄)) |
29 | 1, 15, 18, 24, 12, 6 | ringassd 20275 | . . . . 5 ⊢ (𝜑 → ((((invr‘𝑅)‘𝐼) · 𝐼) · 𝑄) = (((invr‘𝑅)‘𝐼) · (𝐼 · 𝑄))) |
30 | 1, 15, 25, 18, 6 | ringlidmd 20286 | . . . . 5 ⊢ (𝜑 → ((1r‘𝑅) · 𝑄) = 𝑄) |
31 | 28, 29, 30 | 3eqtr3d 2783 | . . . 4 ⊢ (𝜑 → (((invr‘𝑅)‘𝐼) · (𝐼 · 𝑄)) = 𝑄) |
32 | 21, 24, 31 | rspcedvdw 3625 | . . 3 ⊢ (𝜑 → ∃𝑖 ∈ (Base‘𝑅)(𝑖 · (𝐼 · 𝑄)) = 𝑄) |
33 | 1, 3, 15 | dvdsr 20379 | . . 3 ⊢ ((𝐼 · 𝑄)(∥r‘𝑅)𝑄 ↔ ((𝐼 · 𝑄) ∈ (Base‘𝑅) ∧ ∃𝑖 ∈ (Base‘𝑅)(𝑖 · (𝐼 · 𝑄)) = 𝑄)) |
34 | 19, 32, 33 | sylanbrc 583 | . 2 ⊢ (𝜑 → (𝐼 · 𝑄)(∥r‘𝑅)𝑄) |
35 | 1, 2, 3, 4, 5, 17, 34 | rprmasso 33533 | 1 ⊢ (𝜑 → (𝐼 · 𝑄) ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 1rcur 20199 Ringcrg 20251 ∥rcdsr 20371 Unitcui 20372 invrcinvr 20404 RPrimecrpm 20449 IDomncidom 20710 |
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-rep 5285 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-int 4952 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-1st 8013 df-2nd 8014 df-tpos 8250 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-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-rprm 20450 df-subrg 20587 df-idom 20713 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-prmidl 33444 |
This theorem is referenced by: 1arithufdlem3 33554 |
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