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Mirrors > Home > MPE Home > Th. List > rngqiprnglinlem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for rngqiprnglin 21065. (Contributed by AV, 28-Feb-2025.) (Proof shortened by AV, 24-Mar-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) |
Ref | Expression |
---|---|
rngqiprnglinlem1 | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · ( 1 · 𝐶)) = ( 1 · (𝐴 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rng2idlring.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
2 | 1 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐼 ∈ (2Ideal‘𝑅)) |
3 | rng2idlring.j | . . . . . . 7 ⊢ 𝐽 = (𝑅 ↾s 𝐼) | |
4 | rng2idlring.t | . . . . . . 7 ⊢ · = (.r‘𝑅) | |
5 | 3, 4 | ressmulr 17259 | . . . . . 6 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽)) |
6 | 2, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → · = (.r‘𝐽)) |
7 | 6 | oveqd 7429 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 1 ) = (( 1 · 𝐴)(.r‘𝐽) 1 )) |
8 | eqid 2731 | . . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
9 | eqid 2731 | . . . . 5 ⊢ (.r‘𝐽) = (.r‘𝐽) | |
10 | rng2idlring.1 | . . . . 5 ⊢ 1 = (1r‘𝐽) | |
11 | rng2idlring.u | . . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Ring) | |
12 | 11 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐽 ∈ Ring) |
13 | rng2idlring.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Rng) | |
14 | rng2idlring.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑅) | |
15 | 13, 1, 3, 11, 14, 4, 10 | rngqiprngghmlem1 21050 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ (Base‘𝐽)) |
16 | 15 | adantrr 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · 𝐴) ∈ (Base‘𝐽)) |
17 | 8, 9, 10, 12, 16 | ringridmd 20165 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴)(.r‘𝐽) 1 ) = ( 1 · 𝐴)) |
18 | 7, 17 | eqtrd 2771 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 1 ) = ( 1 · 𝐴)) |
19 | 18 | oveq1d 7427 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ((( 1 · 𝐴) · 1 ) · 𝐶) = (( 1 · 𝐴) · 𝐶)) |
20 | 13 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝑅 ∈ Rng) |
21 | 13, 1, 3, 11, 14, 4, 10 | rngqiprng1elbas 21049 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐵) |
22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 1 ∈ 𝐵) |
23 | simprl 768 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐴 ∈ 𝐵) | |
24 | 14, 4 | rngcl 20062 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ 𝐵) |
25 | 20, 22, 23, 24 | syl3anc 1370 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · 𝐴) ∈ 𝐵) |
26 | simprr 770 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐶 ∈ 𝐵) | |
27 | 14, 4 | rngass 20057 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ (( 1 · 𝐴) ∈ 𝐵 ∧ 1 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ((( 1 · 𝐴) · 1 ) · 𝐶) = (( 1 · 𝐴) · ( 1 · 𝐶))) |
28 | 20, 25, 22, 26, 27 | syl13anc 1371 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ((( 1 · 𝐴) · 1 ) · 𝐶) = (( 1 · 𝐴) · ( 1 · 𝐶))) |
29 | 14, 4 | rngass 20057 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ ( 1 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 𝐶) = ( 1 · (𝐴 · 𝐶))) |
30 | 20, 22, 23, 26, 29 | syl13anc 1371 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 𝐶) = ( 1 · (𝐴 · 𝐶))) |
31 | 19, 28, 30 | 3eqtr3d 2779 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · ( 1 · 𝐶)) = ( 1 · (𝐴 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 .rcmulr 17205 Rngcrng 20050 1rcur 20079 Ringcrg 20131 2Idealc2idl 21009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-subg 19043 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-subrng 20438 df-lss 20691 df-sra 20934 df-rgmod 20935 df-lidl 20936 df-2idl 21010 |
This theorem is referenced by: rngqiprnglin 21065 |
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