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Mathbox for Thierry Arnoux |
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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 6915 | . 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 20143 | . . 3 ⊢ 𝐵 = (Base‘𝐺) |
7 | elringlsm.2 | . . . 4 ⊢ · = (.r‘𝑅) | |
8 | 1, 7 | mgpplusg 20141 | . . 3 ⊢ · = (+g‘𝐺) |
9 | elringlsm.4 | . . 3 ⊢ × = (LSSum‘𝐺) | |
10 | 6, 8, 9 | lsmelvalx 19658 | . 2 ⊢ ((𝐺 ∈ V ∧ 𝐸 ⊆ 𝐵 ∧ 𝐹 ⊆ 𝐵) → (𝑍 ∈ (𝐸 × 𝐹) ↔ ∃𝑥 ∈ 𝐸 ∃𝑦 ∈ 𝐹 𝑍 = (𝑥 · 𝑦))) |
11 | 2, 3, 4, 10 | mp3an2i 1464 | 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 1535 ∈ wcel 2104 ∃wrex 3066 Vcvv 3477 ⊆ wss 3963 ‘cfv 6558 (class class class)co 7425 Basecbs 17234 .rcmulr 17288 LSSumclsm 19652 mulGrpcmgp 20137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-nn 12258 df-2 12320 df-sets 17187 df-slot 17205 df-ndx 17217 df-base 17235 df-plusg 17300 df-lsm 19654 df-mgp 20138 |
This theorem is referenced by: elringlsmd 33365 ringlsmss1 33367 ringlsmss2 33368 |
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