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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elringlsmd | Structured version Visualization version GIF version |
Description: Membership in a product of two subsets of a ring, one direction. (Contributed by Thierry Arnoux, 13-Apr-2024.) |
Ref | Expression |
---|---|
elringlsm.1 | ⊢ 𝐵 = (Base‘𝑅) |
elringlsm.2 | ⊢ · = (.r‘𝑅) |
elringlsm.3 | ⊢ 𝐺 = (mulGrp‘𝑅) |
elringlsm.4 | ⊢ × = (LSSum‘𝐺) |
elringlsm.6 | ⊢ (𝜑 → 𝐸 ⊆ 𝐵) |
elringlsm.7 | ⊢ (𝜑 → 𝐹 ⊆ 𝐵) |
elringlsmd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐸) |
elringlsmd.2 | ⊢ (𝜑 → 𝑌 ∈ 𝐹) |
Ref | Expression |
---|---|
elringlsmd | ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elringlsmd.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐸) | |
2 | elringlsmd.2 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐹) | |
3 | eqidd 2760 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑌)) | |
4 | rspceov 7190 | . . 3 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐹 ∧ (𝑋 · 𝑌) = (𝑋 · 𝑌)) → ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 (𝑋 · 𝑌) = (𝑥 · 𝑦)) | |
5 | 1, 2, 3, 4 | syl3anc 1369 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 (𝑋 · 𝑌) = (𝑥 · 𝑦)) |
6 | elringlsm.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
7 | elringlsm.2 | . . 3 ⊢ · = (.r‘𝑅) | |
8 | elringlsm.3 | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
9 | elringlsm.4 | . . 3 ⊢ × = (LSSum‘𝐺) | |
10 | elringlsm.6 | . . 3 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
11 | elringlsm.7 | . . 3 ⊢ (𝜑 → 𝐹 ⊆ 𝐵) | |
12 | 6, 7, 8, 9, 10, 11 | elringlsm 31087 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 (𝑋 · 𝑌) = (𝑥 · 𝑦))) |
13 | 5, 12 | mpbird 260 | 1 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ (𝐸 × 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 ∃wrex 3069 ⊆ wss 3854 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 .rcmulr 16609 LSSumclsm 18811 mulGrpcmgp 19292 |
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-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-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 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-ndx 16529 df-slot 16530 df-base 16532 df-sets 16533 df-plusg 16621 df-lsm 18813 df-mgp 19293 |
This theorem is referenced by: idlmulssprm 31123 |
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