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Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmirredlem | Structured version Visualization version GIF version |
Description: Lemma for rprmirred 33538. (Contributed by Thierry Arnoux, 18-May-2025.) |
Ref | Expression |
---|---|
rprmirredlem.1 | ⊢ 𝐵 = (Base‘𝑅) |
rprmirredlem.2 | ⊢ 𝑈 = (Unit‘𝑅) |
rprmirredlem.3 | ⊢ 0 = (0g‘𝑅) |
rprmirredlem.4 | ⊢ · = (.r‘𝑅) |
rprmirredlem.5 | ⊢ ∥ = (∥r‘𝑅) |
rprmirredlem.6 | ⊢ (𝜑 → 𝑅 ∈ IDomn) |
rprmirredlem.7 | ⊢ (𝜑 → 𝑄 ≠ 0 ) |
rprmirredlem.8 | ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ 𝑈)) |
rprmirredlem.9 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
rprmirredlem.10 | ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌)) |
rprmirredlem.11 | ⊢ (𝜑 → 𝑄 ∥ 𝑋) |
Ref | Expression |
---|---|
rprmirredlem | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rprmirredlem.6 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ IDomn) | |
2 | 1 | idomcringd 20743 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ CRing) |
3 | 2 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑅 ∈ CRing) |
4 | rprmirredlem.9 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
5 | 4 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑌 ∈ 𝐵) |
6 | rprmirredlem.1 | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
7 | rprmirredlem.3 | . . . . . . 7 ⊢ 0 = (0g‘𝑅) | |
8 | rprmirredlem.4 | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
9 | 3 | crngringd 20263 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑅 ∈ Ring) |
10 | simplr 769 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑡 ∈ 𝐵) | |
11 | 6, 8, 9, 10, 5 | ringcld 20276 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → (𝑡 · 𝑌) ∈ 𝐵) |
12 | eqid 2734 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
13 | 6, 12 | ringidcl 20279 | . . . . . . . 8 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
14 | 9, 13 | syl 17 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → (1r‘𝑅) ∈ 𝐵) |
15 | rprmirredlem.11 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∥ 𝑋) | |
16 | rprmirredlem.5 | . . . . . . . . . . . 12 ⊢ ∥ = (∥r‘𝑅) | |
17 | 6, 16, 8 | dvdsr 20378 | . . . . . . . . . . 11 ⊢ (𝑄 ∥ 𝑋 ↔ (𝑄 ∈ 𝐵 ∧ ∃𝑡 ∈ 𝐵 (𝑡 · 𝑄) = 𝑋)) |
18 | 15, 17 | sylib 218 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑄 ∈ 𝐵 ∧ ∃𝑡 ∈ 𝐵 (𝑡 · 𝑄) = 𝑋)) |
19 | 18 | simpld 494 | . . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐵) |
20 | 19 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑄 ∈ 𝐵) |
21 | rprmirredlem.7 | . . . . . . . . 9 ⊢ (𝜑 → 𝑄 ≠ 0 ) | |
22 | 21 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑄 ≠ 0 ) |
23 | 20, 22 | eldifsnd 4791 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑄 ∈ (𝐵 ∖ { 0 })) |
24 | 1 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑅 ∈ IDomn) |
25 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → (𝑡 · 𝑄) = 𝑋) | |
26 | 25 | oveq1d 7445 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑄) · 𝑌) = (𝑋 · 𝑌)) |
27 | rprmirredlem.10 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑄 = (𝑋 · 𝑌)) | |
28 | 27 | ad2antrr 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑄 = (𝑋 · 𝑌)) |
29 | 26, 28 | eqtr4d 2777 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑄) · 𝑌) = 𝑄) |
30 | 6, 8, 3, 10, 5, 20 | cringmul32d 33217 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑌) · 𝑄) = ((𝑡 · 𝑄) · 𝑌)) |
31 | 6, 8, 12, 9, 20 | ringlidmd 20285 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((1r‘𝑅) · 𝑄) = 𝑄) |
32 | 29, 30, 31 | 3eqtr4d 2784 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑌) · 𝑄) = ((1r‘𝑅) · 𝑄)) |
33 | 6, 7, 8, 11, 14, 23, 24, 32 | idomrcan 33262 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → (𝑡 · 𝑌) = (1r‘𝑅)) |
34 | 18 | simprd 495 | . . . . . 6 ⊢ (𝜑 → ∃𝑡 ∈ 𝐵 (𝑡 · 𝑄) = 𝑋) |
35 | 33, 34 | reximddv3 3169 | . . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐵 (𝑡 · 𝑌) = (1r‘𝑅)) |
36 | 35 | ad2antrr 726 | . . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ∃𝑡 ∈ 𝐵 (𝑡 · 𝑌) = (1r‘𝑅)) |
37 | 6, 16, 8 | dvdsr 20378 | . . . 4 ⊢ (𝑌 ∥ (1r‘𝑅) ↔ (𝑌 ∈ 𝐵 ∧ ∃𝑡 ∈ 𝐵 (𝑡 · 𝑌) = (1r‘𝑅))) |
38 | 5, 36, 37 | sylanbrc 583 | . . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑌 ∥ (1r‘𝑅)) |
39 | rprmirredlem.2 | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
40 | 39, 12, 16 | crngunit 20394 | . . . 4 ⊢ (𝑅 ∈ CRing → (𝑌 ∈ 𝑈 ↔ 𝑌 ∥ (1r‘𝑅))) |
41 | 40 | biimpar 477 | . . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑌 ∥ (1r‘𝑅)) → 𝑌 ∈ 𝑈) |
42 | 3, 38, 41 | syl2anc 584 | . 2 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → 𝑌 ∈ 𝑈) |
43 | 42, 34 | r19.29a 3159 | 1 ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∖ cdif 3959 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 .rcmulr 17298 0gc0g 17485 1rcur 20198 Ringcrg 20250 CRingccrg 20251 ∥rcdsr 20370 Unitcui 20371 IDomncidom 20709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966 df-minusg 18967 df-sbg 18968 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-nzr 20529 df-domn 20711 df-idom 20712 |
This theorem is referenced by: rprmirred 33538 |
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