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Mirrors > Home > MPE Home > Th. List > rngqipring1 | Structured version Visualization version GIF version |
Description: The ring unity of the product of the quotient with a two-sided ideal and the two-sided ideal, which both are rings. (Contributed by AV, 16-Mar-2025.) |
Ref | Expression |
---|---|
rngqiprngfu.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
rngqiprngfu.i | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
rngqiprngfu.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) |
rngqiprngfu.u | ⊢ (𝜑 → 𝐽 ∈ Ring) |
rngqiprngfu.b | ⊢ 𝐵 = (Base‘𝑅) |
rngqiprngfu.t | ⊢ · = (.r‘𝑅) |
rngqiprngfu.1 | ⊢ 1 = (1r‘𝐽) |
rngqiprngfu.g | ⊢ ∼ = (𝑅 ~QG 𝐼) |
rngqiprngfu.q | ⊢ 𝑄 = (𝑅 /s ∼ ) |
rngqiprngfu.v | ⊢ (𝜑 → 𝑄 ∈ Ring) |
rngqiprngfu.e | ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) |
rngqiprngfu.m | ⊢ − = (-g‘𝑅) |
rngqiprngfu.a | ⊢ + = (+g‘𝑅) |
rngqiprngfu.n | ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 ) |
rngqipring1.p | ⊢ 𝑃 = (𝑄 ×s 𝐽) |
Ref | Expression |
---|---|
rngqipring1 | ⊢ (𝜑 → (1r‘𝑃) = 〈[𝐸] ∼ , 1 〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngqipring1.p | . . 3 ⊢ 𝑃 = (𝑄 ×s 𝐽) | |
2 | rngqiprngfu.v | . . 3 ⊢ (𝜑 → 𝑄 ∈ Ring) | |
3 | rngqiprngfu.u | . . 3 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
4 | 1, 2, 3 | xpsring1d 20347 | . 2 ⊢ (𝜑 → (1r‘𝑃) = 〈(1r‘𝑄), (1r‘𝐽)〉) |
5 | rngqiprngfu.e | . . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) | |
6 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (1r‘𝑄)) |
7 | eleq2 2828 | . . . . . . . . . . 11 ⊢ ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ )) | |
8 | 7 | adantl 481 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) ↔ 𝐸 ∈ [𝑥] ∼ )) |
9 | elecg 8788 | . . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ (1r‘𝑄) ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸)) | |
10 | 5, 9 | sylan 580 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ ↔ 𝑥 ∼ 𝐸)) |
11 | rngqiprngfu.r | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
12 | rngqiprngfu.i | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
13 | rngqiprngfu.j | . . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
14 | ringrng 20299 | . . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng) | |
15 | 3, 14 | syl 17 | . . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐽 ∈ Rng) |
16 | 13, 15 | eqeltrrid 2844 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng) |
17 | 11, 12, 16 | rng2idlnsg 21294 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) |
18 | nsgsubg 19189 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
19 | 17, 18 | syl 17 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅)) |
20 | 19 | adantr 480 | . . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼 ∈ (SubGrp‘𝑅)) |
21 | rngqiprngfu.b | . . . . . . . . . . . . . . . . . 18 ⊢ 𝐵 = (Base‘𝑅) | |
22 | rngqiprngfu.g | . . . . . . . . . . . . . . . . . 18 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
23 | 21, 22 | eqger 19209 | . . . . . . . . . . . . . . . . 17 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → ∼ Er 𝐵) |
24 | 20, 23 | syl 17 | . . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ Er 𝐵) |
25 | simpr 484 | . . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
26 | 24, 25 | erth 8795 | . . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 ↔ [𝑥] ∼ = [𝐸] ∼ )) |
27 | 26 | biimpa 476 | . . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝑥] ∼ = [𝐸] ∼ ) |
28 | 27 | eqcomd 2741 | . . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑥 ∼ 𝐸) → [𝐸] ∼ = [𝑥] ∼ ) |
29 | 28 | ex 412 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∼ 𝐸 → [𝐸] ∼ = [𝑥] ∼ )) |
30 | 10, 29 | sylbid 240 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) |
31 | 30 | adantr 480 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) |
32 | 8, 31 | sylbid 240 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ )) |
33 | 32 | ex 412 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → (𝐸 ∈ (1r‘𝑄) → [𝐸] ∼ = [𝑥] ∼ ))) |
34 | 6, 33 | mpid 44 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((1r‘𝑄) = [𝑥] ∼ → [𝐸] ∼ = [𝑥] ∼ )) |
35 | 34 | imp 406 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = [𝑥] ∼ ) |
36 | simpr 484 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → (1r‘𝑄) = [𝑥] ∼ ) | |
37 | 35, 36 | eqtr4d 2778 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ (1r‘𝑄) = [𝑥] ∼ ) → [𝐸] ∼ = (1r‘𝑄)) |
38 | rngqiprngfu.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
39 | rngqiprngfu.1 | . . . . . 6 ⊢ 1 = (1r‘𝐽) | |
40 | rngqiprngfu.q | . . . . . 6 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
41 | 11, 12, 13, 3, 21, 38, 39, 22, 40, 2 | rngqiprngfulem1 21339 | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (1r‘𝑄) = [𝑥] ∼ ) |
42 | 37, 41 | r19.29a 3160 | . . . 4 ⊢ (𝜑 → [𝐸] ∼ = (1r‘𝑄)) |
43 | 42 | eqcomd 2741 | . . 3 ⊢ (𝜑 → (1r‘𝑄) = [𝐸] ∼ ) |
44 | 39 | eqcomi 2744 | . . . 4 ⊢ (1r‘𝐽) = 1 |
45 | 44 | a1i 11 | . . 3 ⊢ (𝜑 → (1r‘𝐽) = 1 ) |
46 | 43, 45 | opeq12d 4886 | . 2 ⊢ (𝜑 → 〈(1r‘𝑄), (1r‘𝐽)〉 = 〈[𝐸] ∼ , 1 〉) |
47 | 4, 46 | eqtrd 2775 | 1 ⊢ (𝜑 → (1r‘𝑃) = 〈[𝐸] ∼ , 1 〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Er wer 8741 [cec 8742 Basecbs 17245 ↾s cress 17274 +gcplusg 17298 .rcmulr 17299 /s cqus 17552 ×s cxps 17553 -gcsg 18966 SubGrpcsubg 19151 NrmSGrpcnsg 19152 ~QG cqg 19153 Rngcrng 20170 1rcur 20199 Ringcrg 20251 2Idealc2idl 21277 |
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-tp 4636 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-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 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-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 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-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-imas 17555 df-qus 17556 df-xps 17557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-nsg 19155 df-eqg 19156 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-subrng 20563 df-lss 20948 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-2idl 21278 |
This theorem is referenced by: rngqiprngu 21346 |
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