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| Mirrors > Home > MPE Home > Th. List > rngqiprnglinlem1 | Structured version Visualization version GIF version | ||
| Description: Lemma 1 for rngqiprnglin 21312. (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 17351 | . . . . . 6 ⊢ (𝐼 ∈ (2Ideal‘𝑅) → · = (.r‘𝐽)) |
| 6 | 2, 5 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → · = (.r‘𝐽)) |
| 7 | 6 | oveqd 7448 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 1 ) = (( 1 · 𝐴)(.r‘𝐽) 1 )) |
| 8 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐽) = (Base‘𝐽) | |
| 9 | eqid 2737 | . . . . 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 21297 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ (Base‘𝐽)) |
| 16 | 15 | adantrr 717 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · 𝐴) ∈ (Base‘𝐽)) |
| 17 | 8, 9, 10, 12, 16 | ringridmd 20270 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴)(.r‘𝐽) 1 ) = ( 1 · 𝐴)) |
| 18 | 7, 17 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 1 ) = ( 1 · 𝐴)) |
| 19 | 18 | oveq1d 7446 | . 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 21296 | . . . . 5 ⊢ (𝜑 → 1 ∈ 𝐵) |
| 22 | 21 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 1 ∈ 𝐵) |
| 23 | simprl 771 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐴 ∈ 𝐵) | |
| 24 | 14, 4 | rngcl 20161 | . . . 4 ⊢ ((𝑅 ∈ Rng ∧ 1 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵) → ( 1 · 𝐴) ∈ 𝐵) |
| 25 | 20, 22, 23, 24 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ( 1 · 𝐴) ∈ 𝐵) |
| 26 | simprr 773 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → 𝐶 ∈ 𝐵) | |
| 27 | 14, 4 | rngass 20156 | . . 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 1374 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → ((( 1 · 𝐴) · 1 ) · 𝐶) = (( 1 · 𝐴) · ( 1 · 𝐶))) |
| 29 | 14, 4 | rngass 20156 | . . 3 ⊢ ((𝑅 ∈ Rng ∧ ( 1 ∈ 𝐵 ∧ 𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 𝐶) = ( 1 · (𝐴 · 𝐶))) |
| 30 | 20, 22, 23, 26, 29 | syl13anc 1374 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝐵 ∧ 𝐶 ∈ 𝐵)) → (( 1 · 𝐴) · 𝐶) = ( 1 · (𝐴 · 𝐶))) |
| 31 | 19, 28, 30 | 3eqtr3d 2785 | 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 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 .rcmulr 17298 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-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-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-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-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-subg 19141 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: rngqiprnglin 21312 |
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