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| Mirrors > Home > MPE Home > Th. List > Mathboxes > elringlsm | Structured version Visualization version GIF version | ||
| Description: Membership in a product of two subsets of a ring. (Contributed by Thierry Arnoux, 20-Jan-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) |
| Ref | Expression |
|---|---|
| elringlsm | ⊢ (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elringlsm.3 | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
| 2 | 1 | fvexi 6918 | . 2 ⊢ 𝐺 ∈ V |
| 3 | elringlsm.6 | . 2 ⊢ (𝜑 → 𝐸 ⊆ 𝐵) | |
| 4 | elringlsm.7 | . 2 ⊢ (𝜑 → 𝐹 ⊆ 𝐵) | |
| 5 | elringlsm.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | 1, 5 | mgpbas 20138 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
| 7 | elringlsm.2 | . . . 4 ⊢ · = (.r‘𝑅) | |
| 8 | 1, 7 | mgpplusg 20137 | . . 3 ⊢ · = (+g‘𝐺) |
| 9 | elringlsm.4 | . . 3 ⊢ × = (LSSum‘𝐺) | |
| 10 | 6, 8, 9 | lsmelvalx 19654 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐸 ⊆ 𝐵 ∧ 𝐹 ⊆ 𝐵) → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦))) |
| 11 | 2, 3, 4, 10 | mp3an2i 1468 | 1 ⊢ (𝜑 → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3069 Vcvv 3479 ⊆ wss 3950 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 .rcmulr 17294 LSSumclsm 19648 mulGrpcmgp 20133 |
| 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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
| 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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-plusg 17306 df-lsm 19650 df-mgp 20134 |
| This theorem is referenced by: elringlsmd 33409 ringlsmss1 33411 ringlsmss2 33412 |
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