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| Mirrors > Home > MPE Home > Th. List > rngqiprngfulem4 | Structured version Visualization version GIF version | ||
| Description: Lemma 4 for rngqiprngfu 21327. (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 ) |
| Ref | Expression |
|---|---|
| rngqiprngfulem4 | ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rngqiprngfu.n | . . . . . 6 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 ) | |
| 2 | 1 | oveq2i 7442 | . . . . 5 ⊢ (𝐸 − 𝑈) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 )) |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐸 − 𝑈) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 ))) |
| 4 | rngqiprngfu.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
| 5 | rngqiprngfu.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
| 6 | rngqiprngfu.m | . . . . 5 ⊢ − = (-g‘𝑅) | |
| 7 | rngqiprngfu.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
| 8 | rngabl 20152 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel) | |
| 9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Abel) |
| 10 | rngqiprngfu.i | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
| 11 | rngqiprngfu.j | . . . . . 6 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
| 12 | rngqiprngfu.u | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
| 13 | rngqiprngfu.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
| 14 | rngqiprngfu.1 | . . . . . 6 ⊢ 1 = (1r‘𝐽) | |
| 15 | rngqiprngfu.g | . . . . . 6 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
| 16 | rngqiprngfu.q | . . . . . 6 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
| 17 | rngqiprngfu.v | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Ring) | |
| 18 | rngqiprngfu.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) | |
| 19 | 7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18 | rngqiprngfulem2 21322 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵) |
| 20 | rnggrp 20155 | . . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
| 21 | 7, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp) |
| 22 | 7, 10, 11, 12, 4, 13, 14 | rngqiprng1elbas 21296 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ 𝐵) |
| 23 | 4, 13 | rngcl 20161 | . . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵) |
| 24 | 7, 22, 19, 23 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵) |
| 25 | 4, 6 | grpsubcl 19038 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵) |
| 26 | 21, 19, 24, 25 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵) |
| 27 | 4, 5, 6, 9, 19, 26, 22 | ablsubsub4 19836 | . . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (𝐸 − ((𝐸 − ( 1 · 𝐸)) + 1 ))) |
| 28 | 4, 6, 9, 19, 24 | ablnncan 19838 | . . . . 5 ⊢ (𝜑 → (𝐸 − (𝐸 − ( 1 · 𝐸))) = ( 1 · 𝐸)) |
| 29 | 28 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → ((𝐸 − (𝐸 − ( 1 · 𝐸))) − 1 ) = (( 1 · 𝐸) − 1 )) |
| 30 | 3, 27, 29 | 3eqtr2d 2783 | . . 3 ⊢ (𝜑 → (𝐸 − 𝑈) = (( 1 · 𝐸) − 1 )) |
| 31 | ringrng 20282 | . . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 𝐽 ∈ Rng) | |
| 32 | 12, 31 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Rng) |
| 33 | 11, 32 | eqeltrrid 2846 | . . . . . . . 8 ⊢ (𝜑 → (𝑅 ↾s 𝐼) ∈ Rng) |
| 34 | 7, 10, 33 | rng2idlnsg 21276 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (NrmSGrp‘𝑅)) |
| 35 | nsgsubg 19176 | . . . . . . 7 ⊢ (𝐼 ∈ (NrmSGrp‘𝑅) → 𝐼 ∈ (SubGrp‘𝑅)) | |
| 36 | 34, 35 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅)) |
| 37 | 7, 10, 11, 12, 4, 13, 14 | rngqiprngghmlem1 21297 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ (Base‘𝐽)) |
| 38 | 19, 37 | mpdan 687 | . . . . . . 7 ⊢ (𝜑 → ( 1 · 𝐸) ∈ (Base‘𝐽)) |
| 39 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
| 40 | 10, 11, 39 | 2idlbas 21273 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐽) = 𝐼) |
| 41 | 38, 40 | eleqtrd 2843 | . . . . . 6 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐼) |
| 42 | 39, 14 | ringidcl 20262 | . . . . . . . 8 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽)) |
| 43 | 12, 42 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ (Base‘𝐽)) |
| 44 | 43, 40 | eleqtrd 2843 | . . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐼) |
| 45 | eqid 2737 | . . . . . . 7 ⊢ (-g‘𝐽) = (-g‘𝐽) | |
| 46 | 6, 11, 45 | subgsub 19156 | . . . . . 6 ⊢ ((𝐼 ∈ (SubGrp‘𝑅) ∧ ( 1 · 𝐸) ∈ 𝐼 ∧ 1 ∈ 𝐼) → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-g‘𝐽) 1 )) |
| 47 | 36, 41, 44, 46 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) = (( 1 · 𝐸)(-g‘𝐽) 1 )) |
| 48 | 12 | ringgrpd 20239 | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Grp) |
| 49 | 39, 45 | grpsubcl 19038 | . . . . . 6 ⊢ ((𝐽 ∈ Grp ∧ ( 1 · 𝐸) ∈ (Base‘𝐽) ∧ 1 ∈ (Base‘𝐽)) → (( 1 · 𝐸)(-g‘𝐽) 1 ) ∈ (Base‘𝐽)) |
| 50 | 48, 38, 43, 49 | syl3anc 1373 | . . . . 5 ⊢ (𝜑 → (( 1 · 𝐸)(-g‘𝐽) 1 ) ∈ (Base‘𝐽)) |
| 51 | 47, 50 | eqeltrd 2841 | . . . 4 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ (Base‘𝐽)) |
| 52 | 51, 40 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → (( 1 · 𝐸) − 1 ) ∈ 𝐼) |
| 53 | 30, 52 | eqeltrd 2841 | . 2 ⊢ (𝜑 → (𝐸 − 𝑈) ∈ 𝐼) |
| 54 | 7, 10, 11, 12, 4, 13, 14, 15, 16, 17, 18, 6, 5, 1 | rngqiprngfulem3 21323 | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐵) |
| 55 | 4, 6, 15 | qusecsub 19853 | . . 3 ⊢ (((𝑅 ∈ Abel ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑈 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼)) |
| 56 | 9, 36, 54, 19, 55 | syl22anc 839 | . 2 ⊢ (𝜑 → ([𝑈] ∼ = [𝐸] ∼ ↔ (𝐸 − 𝑈) ∈ 𝐼)) |
| 57 | 53, 56 | mpbird 257 | 1 ⊢ (𝜑 → [𝑈] ∼ = [𝐸] ∼ ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 [cec 8743 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 .rcmulr 17298 /s cqus 17550 Grpcgrp 18951 -gcsg 18953 SubGrpcsubg 19138 NrmSGrpcnsg 19139 ~QG cqg 19140 Abelcabl 19799 Rngcrng 20149 1rcur 20178 Ringcrg 20230 2Idealc2idl 21259 |
| 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-tp 4631 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-1o 8506 df-er 8745 df-ec 8747 df-qs 8751 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 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-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-subrng 20546 df-lss 20930 df-sra 21172 df-rgmod 21173 df-lidl 21218 df-2idl 21260 |
| This theorem is referenced by: rngqiprngfu 21327 |
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