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Mirrors > Home > MPE Home > Th. List > rngqiprngfulem5 | Structured version Visualization version GIF version |
Description: Lemma 5 for rngqiprngfu 21200. (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 |
---|---|
rngqiprngfulem5 | ⊢ (𝜑 → ( 1 · 𝑈) = 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngqiprngfu.n | . . . 4 ⊢ 𝑈 = ((𝐸 − ( 1 · 𝐸)) + 1 ) | |
2 | 1 | oveq2i 7425 | . . 3 ⊢ ( 1 · 𝑈) = ( 1 · ((𝐸 − ( 1 · 𝐸)) + 1 )) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → ( 1 · 𝑈) = ( 1 · ((𝐸 − ( 1 · 𝐸)) + 1 ))) |
4 | rngqiprngfu.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
5 | rngqiprngfu.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
6 | rngqiprngfu.j | . . . . 5 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
7 | rngqiprngfu.u | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
8 | rngqiprngfu.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
9 | rngqiprngfu.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
10 | rngqiprngfu.1 | . . . . 5 ⊢ 1 = (1r‘𝐽) | |
11 | 4, 5, 6, 7, 8, 9, 10 | rngqiprng1elbas 21169 | . . . 4 ⊢ (𝜑 → 1 ∈ 𝐵) |
12 | rnggrp 20091 | . . . . . 6 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
13 | 4, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) |
14 | rngqiprngfu.g | . . . . . 6 ⊢ ∼ = (𝑅 ~QG 𝐼) | |
15 | rngqiprngfu.q | . . . . . 6 ⊢ 𝑄 = (𝑅 /s ∼ ) | |
16 | rngqiprngfu.v | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ Ring) | |
17 | rngqiprngfu.e | . . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1r‘𝑄)) | |
18 | 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17 | rngqiprngfulem2 21195 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐵) |
19 | 8, 9 | rngcl 20097 | . . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → ( 1 · 𝐸) ∈ 𝐵) |
20 | 4, 11, 18, 19 | syl3anc 1369 | . . . . 5 ⊢ (𝜑 → ( 1 · 𝐸) ∈ 𝐵) |
21 | rngqiprngfu.m | . . . . . 6 ⊢ − = (-g‘𝑅) | |
22 | 8, 21 | grpsubcl 18969 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐸 ∈ 𝐵 ∧ ( 1 · 𝐸) ∈ 𝐵) → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵) |
23 | 13, 18, 20, 22 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝐸 − ( 1 · 𝐸)) ∈ 𝐵) |
24 | rngqiprngfu.a | . . . . 5 ⊢ + = (+g‘𝑅) | |
25 | 8, 24, 9 | rngdi 20093 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ ( 1 ∈ 𝐵 ∧ (𝐸 − ( 1 · 𝐸)) ∈ 𝐵 ∧ 1 ∈ 𝐵)) → ( 1 · ((𝐸 − ( 1 · 𝐸)) + 1 )) = (( 1 · (𝐸 − ( 1 · 𝐸))) + ( 1 · 1 ))) |
26 | 4, 11, 23, 11, 25 | syl13anc 1370 | . . 3 ⊢ (𝜑 → ( 1 · ((𝐸 − ( 1 · 𝐸)) + 1 )) = (( 1 · (𝐸 − ( 1 · 𝐸))) + ( 1 · 1 ))) |
27 | 8, 9, 21, 4, 11, 18, 20 | rngsubdi 20104 | . . . . 5 ⊢ (𝜑 → ( 1 · (𝐸 − ( 1 · 𝐸))) = (( 1 · 𝐸) − ( 1 · ( 1 · 𝐸)))) |
28 | 8, 9 | rngass 20092 | . . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ ( 1 ∈ 𝐵 ∧ 1 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵)) → (( 1 · 1 ) · 𝐸) = ( 1 · ( 1 · 𝐸))) |
29 | 4, 11, 11, 18, 28 | syl13anc 1370 | . . . . . . 7 ⊢ (𝜑 → (( 1 · 1 ) · 𝐸) = ( 1 · ( 1 · 𝐸))) |
30 | 6, 9 | ressmulr 17281 | . . . . . . . . . . 11 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽)) |
31 | 5, 30 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → · = (.r‘𝐽)) |
32 | 31 | oveqd 7431 | . . . . . . . . 9 ⊢ (𝜑 → ( 1 · 1 ) = ( 1 (.r‘𝐽) 1 )) |
33 | eqid 2727 | . . . . . . . . . . 11 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
34 | 33, 10 | ringidcl 20195 | . . . . . . . . . 10 ⊢ (𝐽 ∈ Ring → 1 ∈ (Base‘𝐽)) |
35 | eqid 2727 | . . . . . . . . . . 11 ⊢ (.r‘𝐽) = (.r‘𝐽) | |
36 | 33, 35, 10 | ringlidm 20198 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ Ring ∧ 1 ∈ (Base‘𝐽)) → ( 1 (.r‘𝐽) 1 ) = 1 ) |
37 | 7, 34, 36 | syl2anc2 584 | . . . . . . . . 9 ⊢ (𝜑 → ( 1 (.r‘𝐽) 1 ) = 1 ) |
38 | 32, 37 | eqtrd 2767 | . . . . . . . 8 ⊢ (𝜑 → ( 1 · 1 ) = 1 ) |
39 | 38 | oveq1d 7429 | . . . . . . 7 ⊢ (𝜑 → (( 1 · 1 ) · 𝐸) = ( 1 · 𝐸)) |
40 | 29, 39 | eqtr3d 2769 | . . . . . 6 ⊢ (𝜑 → ( 1 · ( 1 · 𝐸)) = ( 1 · 𝐸)) |
41 | 40 | oveq2d 7430 | . . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − ( 1 · ( 1 · 𝐸))) = (( 1 · 𝐸) − ( 1 · 𝐸))) |
42 | eqid 2727 | . . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
43 | 8, 42, 21 | grpsubid 18973 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ ( 1 · 𝐸) ∈ 𝐵) → (( 1 · 𝐸) − ( 1 · 𝐸)) = (0g‘𝑅)) |
44 | 13, 20, 43 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (( 1 · 𝐸) − ( 1 · 𝐸)) = (0g‘𝑅)) |
45 | 27, 41, 44 | 3eqtrd 2771 | . . . 4 ⊢ (𝜑 → ( 1 · (𝐸 − ( 1 · 𝐸))) = (0g‘𝑅)) |
46 | 45, 38 | oveq12d 7432 | . . 3 ⊢ (𝜑 → (( 1 · (𝐸 − ( 1 · 𝐸))) + ( 1 · 1 )) = ((0g‘𝑅) + 1 )) |
47 | 26, 46 | eqtrd 2767 | . 2 ⊢ (𝜑 → ( 1 · ((𝐸 − ( 1 · 𝐸)) + 1 )) = ((0g‘𝑅) + 1 )) |
48 | 8, 24, 42, 13, 11 | grplidd 18919 | . 2 ⊢ (𝜑 → ((0g‘𝑅) + 1 ) = 1 ) |
49 | 3, 47, 48 | 3eqtrd 2771 | 1 ⊢ (𝜑 → ( 1 · 𝑈) = 1 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 ↾s cress 17202 +gcplusg 17226 .rcmulr 17227 0gc0g 17414 /s cqus 17480 Grpcgrp 18883 -gcsg 18885 ~QG cqg 19070 Rngcrng 20085 1rcur 20114 Ringcrg 20166 2Idealc2idl 21136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-ec 8720 df-qs 8724 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-0g 17416 df-imas 17483 df-qus 17484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-eqg 19073 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-lss 20809 df-sra 21051 df-rgmod 21052 df-lidl 21097 df-2idl 21137 |
This theorem is referenced by: rngqiprngfu 21200 |
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