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Mirrors > Home > MPE Home > Th. List > Mathboxes > drgextgsum | Structured version Visualization version GIF version |
Description: Group sum in a division ring extension. (Contributed by Thierry Arnoux, 17-Jul-2023.) |
Ref | Expression |
---|---|
drgext.b | ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) |
drgext.1 | ⊢ (𝜑 → 𝐸 ∈ DivRing) |
drgext.2 | ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) |
drgext.f | ⊢ 𝐹 = (𝐸 ↾s 𝑈) |
drgext.3 | ⊢ (𝜑 → 𝐹 ∈ DivRing) |
drgextgsum.1 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
Ref | Expression |
---|---|
drgextgsum | ⊢ (𝜑 → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)) = (𝐵 Σg (𝑖 ∈ 𝑋 ↦ 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drgext.b | . 2 ⊢ 𝐵 = ((subringAlg ‘𝐸)‘𝑈) | |
2 | drgextgsum.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
3 | 2 | mptexd 7095 | . 2 ⊢ (𝜑 → (𝑖 ∈ 𝑋 ↦ 𝑌) ∈ V) |
4 | drgext.1 | . 2 ⊢ (𝜑 → 𝐸 ∈ DivRing) | |
5 | drgext.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ DivRing) | |
6 | drgext.2 | . . 3 ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝐸)) | |
7 | drgext.f | . . . 4 ⊢ 𝐹 = (𝐸 ↾s 𝑈) | |
8 | 1, 7 | sralvec 31669 | . . 3 ⊢ ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec) |
9 | 4, 5, 6, 8 | syl3anc 1370 | . 2 ⊢ (𝜑 → 𝐵 ∈ LVec) |
10 | eqid 2740 | . . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸) | |
11 | 10 | subrgss 20021 | . . 3 ⊢ (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸)) |
12 | 6, 11 | syl 17 | . 2 ⊢ (𝜑 → 𝑈 ⊆ (Base‘𝐸)) |
13 | 1, 3, 4, 9, 12 | gsumsra 31301 | 1 ⊢ (𝜑 → (𝐸 Σg (𝑖 ∈ 𝑋 ↦ 𝑌)) = (𝐵 Σg (𝑖 ∈ 𝑋 ↦ 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 ↦ cmpt 5162 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 ↾s cress 16937 Σg cgsu 17147 DivRingcdr 19987 SubRingcsubrg 20016 LVecclvec 20360 subringAlg csra 20426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-seq 13718 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-ip 16976 df-0g 17148 df-gsum 17149 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-grp 18576 df-subg 18748 df-mgp 19717 df-ur 19734 df-ring 19781 df-subrg 20018 df-lmod 20121 df-lvec 20361 df-sra 20430 |
This theorem is referenced by: fedgmullem1 31704 fedgmullem2 31705 extdg1id 31732 |
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