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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rprmirredlem | Structured version Visualization version GIF version | ||
| Description: Lemma for rprmirred 33559. (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 20727 | . . . 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 20243 | . . . . . . . 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 20257 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → (𝑡 · 𝑌) ∈ 𝐵) |
| 12 | eqid 2737 | . . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 13 | 6, 12 | ringidcl 20262 | . . . . . . . 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 20362 | . . . . . . . . . . 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 4787 | . . . . . . 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 7446 | . . . . . . . . 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 2780 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑄) · 𝑌) = 𝑄) |
| 30 | 6, 8, 3, 10, 5, 20 | crng32d 20256 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑌) · 𝑄) = ((𝑡 · 𝑄) · 𝑌)) |
| 31 | 6, 8, 12, 9, 20 | ringlidmd 20269 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((1r‘𝑅) · 𝑄) = 𝑄) |
| 32 | 29, 30, 31 | 3eqtr4d 2787 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑡 · 𝑄) = 𝑋) → ((𝑡 · 𝑌) · 𝑄) = ((1r‘𝑅) · 𝑄)) |
| 33 | 6, 7, 8, 11, 14, 23, 24, 32 | idomrcan 33282 | . . . . . 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 3172 | . . . . 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 20362 | . . . 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 20378 | . . . 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 3162 | 1 ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 0gc0g 17484 1rcur 20178 Ringcrg 20230 CRingccrg 20231 ∥rcdsr 20354 Unitcui 20355 IDomncidom 20693 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-nzr 20513 df-domn 20695 df-idom 20696 |
| This theorem is referenced by: rprmirred 33559 |
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