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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > idlsrgmulrval | Structured version Visualization version GIF version |
Description: Value of the ring multiplication for the ideals of a ring 𝑅. (Contributed by Thierry Arnoux, 1-Jun-2024.) |
Ref | Expression |
---|---|
idlsrgmulrval.1 | ⊢ 𝑆 = (IDLsrg‘𝑅) |
idlsrgmulrval.2 | ⊢ 𝐵 = (LIdeal‘𝑅) |
idlsrgmulrval.3 | ⊢ ⊗ = (.r‘𝑆) |
idlsrgmulrval.4 | ⊢ 𝐺 = (mulGrp‘𝑅) |
idlsrgmulrval.5 | ⊢ · = (LSSum‘𝐺) |
idlsrgmulrval.6 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
idlsrgmulrval.7 | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
idlsrgmulrval.8 | ⊢ (𝜑 → 𝐽 ∈ 𝐵) |
Ref | Expression |
---|---|
idlsrgmulrval | ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idlsrgmulrval.3 | . . 3 ⊢ ⊗ = (.r‘𝑆) | |
2 | idlsrgmulrval.6 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
3 | idlsrgmulrval.1 | . . . . 5 ⊢ 𝑆 = (IDLsrg‘𝑅) | |
4 | idlsrgmulrval.2 | . . . . 5 ⊢ 𝐵 = (LIdeal‘𝑅) | |
5 | idlsrgmulrval.4 | . . . . 5 ⊢ 𝐺 = (mulGrp‘𝑅) | |
6 | idlsrgmulrval.5 | . . . . 5 ⊢ · = (LSSum‘𝐺) | |
7 | 3, 4, 5, 6 | idlsrgmulr 31158 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑥 · 𝑦))) = (.r‘𝑆)) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑥 · 𝑦))) = (.r‘𝑆)) |
9 | 1, 8 | eqtr4id 2813 | . 2 ⊢ (𝜑 → ⊗ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ((RSpan‘𝑅)‘(𝑥 · 𝑦)))) |
10 | oveq12 7152 | . . . 4 ⊢ ((𝑥 = 𝐼 ∧ 𝑦 = 𝐽) → (𝑥 · 𝑦) = (𝐼 · 𝐽)) | |
11 | 10 | adantl 486 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) → (𝑥 · 𝑦) = (𝐼 · 𝐽)) |
12 | 11 | fveq2d 6655 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)) → ((RSpan‘𝑅)‘(𝑥 · 𝑦)) = ((RSpan‘𝑅)‘(𝐼 · 𝐽))) |
13 | idlsrgmulrval.7 | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
14 | idlsrgmulrval.8 | . 2 ⊢ (𝜑 → 𝐽 ∈ 𝐵) | |
15 | fvexd 6666 | . 2 ⊢ (𝜑 → ((RSpan‘𝑅)‘(𝐼 · 𝐽)) ∈ V) | |
16 | 9, 12, 13, 14, 15 | ovmpod 7290 | 1 ⊢ (𝜑 → (𝐼 ⊗ 𝐽) = ((RSpan‘𝑅)‘(𝐼 · 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 .rcmulr 16609 LSSumclsm 18811 mulGrpcmgp 19292 LIdealclidl 19995 RSpancrsp 19996 IDLsrgcidlsrg 31151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-int 4832 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-1st 7686 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-fin 8524 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-2 11722 df-3 11723 df-4 11724 df-5 11725 df-6 11726 df-7 11727 df-8 11728 df-9 11729 df-n0 11920 df-z 12006 df-dec 12123 df-uz 12268 df-fz 12925 df-struct 16528 df-ndx 16529 df-slot 16530 df-base 16532 df-plusg 16621 df-mulr 16622 df-tset 16627 df-ple 16628 df-idlsrg 31152 |
This theorem is referenced by: idlsrgmulrcl 31161 idlsrgmulrss1 31162 idlsrgmulrss2 31163 zarclsun 31326 |
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