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Mirrors > Home > MPE Home > Th. List > rngqiprngimf | Structured version Visualization version GIF version |
Description: 𝐹 is a function from (the base set of) a non-unital ring to the product of the (base set 𝐶 of the) quotient with a two-sided ideal and the (base set 𝐼 of the) two-sided ideal (because of 2idlbas 21291, (Base‘𝐽) = 𝐼!) (Contributed by AV, 21-Feb-2025.) |
Ref | Expression |
---|---|
rng2idlring.r | ⊢ (𝜑 → 𝑅 ∈ Rng) |
rng2idlring.i | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
rng2idlring.j | ⊢ 𝐽 = (𝑅 ↾s 𝐼) |
rng2idlring.u | ⊢ (𝜑 → 𝐽 ∈ Ring) |
rng2idlring.b | ⊢ 𝐵 = (Base‘𝑅) |
rng2idlring.t | ⊢ · = (.r‘𝑅) |
rng2idlring.1 | ⊢ 1 = (1r‘𝐽) |
rngqiprngim.g | ⊢ ∼ = (𝑅 ~QG 𝐼) |
rngqiprngim.q | ⊢ 𝑄 = (𝑅 /s ∼ ) |
rngqiprngim.c | ⊢ 𝐶 = (Base‘𝑄) |
rngqiprngim.p | ⊢ 𝑃 = (𝑄 ×s 𝐽) |
rngqiprngim.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) |
Ref | Expression |
---|---|
rngqiprngimf | ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngqiprngim.g | . . . . . . 7 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
2 | 1 | ovexi 7465 | . . . . . 6 ⊢ ∼ ∈ V |
3 | 2 | ecelqsi 8812 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → [𝑥] ∼ ∈ (𝐵 / ∼ )) |
4 | 3 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥] ∼ ∈ (𝐵 / ∼ )) |
5 | rngqiprngim.q | . . . . . . 7 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
6 | 5 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑄 = (𝑅 /s ∼ )) |
7 | rng2idlring.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘𝑅)) |
9 | 2 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∼ ∈ V) |
10 | rng2idlring.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Rng) |
12 | 6, 8, 9, 11 | qusbas 17592 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / ∼ ) = (Base‘𝑄)) |
13 | rngqiprngim.c | . . . . 5 ⊢ 𝐶 = (Base‘𝑄) | |
14 | 12, 13 | eqtr4di 2793 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / ∼ ) = 𝐶) |
15 | 4, 14 | eleqtrd 2841 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥] ∼ ∈ 𝐶) |
16 | rng2idlring.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
17 | rng2idlring.j | . . . . . . 7 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
18 | eqid 2735 | . . . . . . 7 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
19 | 16, 17, 18 | 2idlbas 21291 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐽) = 𝐼) |
20 | 16, 17, 18 | 2idlelbas 21292 | . . . . . . 7 ⊢ (𝜑 → ((Base‘𝐽) ∈ (LIdeal‘𝑅) ∧ (Base‘𝐽) ∈ (LIdeal‘(oppr‘𝑅)))) |
21 | 20 | simprd 495 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐽) ∈ (LIdeal‘(oppr‘𝑅))) |
22 | 19, 21 | eqeltrrd 2840 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘(oppr‘𝑅))) |
23 | rng2idlring.u | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
24 | ringrng 20299 | . . . . . . . 8 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Rng) |
26 | 17, 25 | eqeltrrid 2844 | . . . . . 6 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng) |
27 | 10, 16, 26 | rng2idl0 21295 | . . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ 𝐼) |
28 | 10, 22, 27 | 3jca 1127 | . . . 4 ⊢ (𝜑 → (𝑅 ∈ Rng ∧ 𝐼 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘𝑅) ∈ 𝐼)) |
29 | rng2idlring.1 | . . . . . . . 8 ⊢ 1 = (1r‘𝐽) | |
30 | 18, 29 | ringidcl 20280 | . . . . . . 7 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽)) |
31 | 23, 30 | syl 17 | . . . . . 6 ⊢ (𝜑 → 1 ∈ (Base‘𝐽)) |
32 | 31, 19 | eleqtrd 2841 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐼) |
33 | 32 | anim1ci 616 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ 𝐵 ∧ 1 ∈ 𝐼)) |
34 | eqid 2735 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
35 | rng2idlring.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
36 | eqid 2735 | . . . . 5 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
37 | 34, 7, 35, 36 | rngridlmcl 21245 | . . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝐼 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘𝑅) ∈ 𝐼) ∧ (𝑥 ∈ 𝐵 ∧ 1 ∈ 𝐼)) → ( 1 · 𝑥) ∈ 𝐼) |
38 | 28, 33, 37 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 1 · 𝑥) ∈ 𝐼) |
39 | 15, 38 | opelxpd 5728 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 〈[𝑥] ∼ , ( 1 · 𝑥)〉 ∈ (𝐶 × 𝐼)) |
40 | rngqiprngim.f | . 2 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ 〈[𝑥] ∼ , ( 1 · 𝑥)〉) | |
41 | 39, 40 | fmptd 7134 | 1 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐶 × 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 [cec 8742 / cqs 8743 Basecbs 17245 ↾s cress 17274 .rcmulr 17299 0gc0g 17486 /s cqus 17552 ×s cxps 17553 ~QG cqg 19153 Rngcrng 20170 1rcur 20199 Ringcrg 20251 opprcoppr 20350 LIdealclidl 21234 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-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 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-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-subrng 20563 df-lss 20948 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-2idl 21278 |
This theorem is referenced by: rngqiprngghm 21327 rngqiprngimfo 21329 |
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