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Mirrors > Home > MPE Home > Th. List > gsumsubm | Structured version Visualization version GIF version |
Description: Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.) |
Ref | Expression |
---|---|
gsumsubm.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsubm.s | ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) |
gsumsubm.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
gsumsubm.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
Ref | Expression |
---|---|
gsumsubm | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2740 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | gsumsubm.h | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
4 | gsumsubm.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | submrcl 18452 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
7 | gsumsubm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1 | submss 18459 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
9 | 4, 8 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
10 | gsumsubm.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
11 | eqid 2740 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 11 | subm0cl 18461 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑆) |
13 | 4, 12 | syl 17 | . 2 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑆) |
14 | 1, 2, 11 | mndlrid 18415 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐺)) → (((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)) |
15 | 6, 14 | sylan 580 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)) |
16 | 1, 2, 3, 6, 7, 9, 10, 13, 15 | gsumress 18377 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ⊆ wss 3892 ⟶wf 6428 ‘cfv 6432 (class class class)co 7272 Basecbs 16923 ↾s cress 16952 +gcplusg 16973 0gc0g 17161 Σg cgsu 17162 Mndcmnd 18396 SubMndcsubmnd 18440 |
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-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7583 ax-cnex 10938 ax-resscn 10939 ax-1cn 10940 ax-icn 10941 ax-addcl 10942 ax-addrcl 10943 ax-mulcl 10944 ax-mulrcl 10945 ax-mulcom 10946 ax-addass 10947 ax-mulass 10948 ax-distr 10949 ax-i2m1 10950 ax-1ne0 10951 ax-1rid 10952 ax-rnegex 10953 ax-rrecex 10954 ax-cnre 10955 ax-pre-lttri 10956 ax-pre-lttrn 10957 ax-pre-ltadd 10958 ax-pre-mulgt0 10959 |
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 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7229 df-ov 7275 df-oprab 7276 df-mpo 7277 df-om 7708 df-2nd 7826 df-frecs 8089 df-wrecs 8120 df-recs 8194 df-rdg 8233 df-er 8490 df-en 8726 df-dom 8727 df-sdom 8728 df-pnf 11022 df-mnf 11023 df-xr 11024 df-ltxr 11025 df-le 11026 df-sub 11218 df-neg 11219 df-nn 11985 df-2 12047 df-seq 13733 df-sets 16876 df-slot 16894 df-ndx 16906 df-base 16924 df-ress 16953 df-plusg 16986 df-0g 17163 df-gsum 17164 df-mgm 18337 df-sgrp 18386 df-mnd 18397 df-submnd 18442 |
This theorem is referenced by: gsumzsubmcl 19530 resspsrmul 21197 gsumply1subr 21416 tsmssubm 23305 amgmlem 26150 lgseisenlem4 26537 fedgmullem1 31719 sge0tsms 43900 amgmwlem 46485 amgmlemALT 46486 |
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