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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumsubg | Structured version Visualization version GIF version |
Description: The group sum in a subgroup is the same as the group sum. (Contributed by Thierry Arnoux, 28-May-2023.) |
Ref | Expression |
---|---|
gsumsubg.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐵) |
gsumsubg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
gsumsubg.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
gsumsubg.b | ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺)) |
Ref | Expression |
---|---|
gsumsubg | ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2725 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | gsumsubg.1 | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝐵) | |
4 | gsumsubg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ (SubGrp‘𝐺)) | |
5 | 4 | elfvexd 6931 | . 2 ⊢ (𝜑 → 𝐺 ∈ V) |
6 | gsumsubg.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | 1 | subgss 19086 | . . 3 ⊢ (𝐵 ∈ (SubGrp‘𝐺) → 𝐵 ⊆ (Base‘𝐺)) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐺)) |
9 | gsumsubg.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
10 | eqid 2725 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
11 | 10 | subg0cl 19093 | . . 3 ⊢ (𝐵 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝐵) |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝐵) |
13 | subgrcl 19090 | . . . 4 ⊢ (𝐵 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
14 | 4, 13 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
15 | 1, 2, 10 | grplid 18928 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → ((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥) |
16 | 1, 2, 10 | grprid 18929 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥) |
17 | 15, 16 | jca 510 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ (Base‘𝐺)) → (((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)) |
18 | 14, 17 | sylan 578 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)) |
19 | 1, 2, 3, 5, 6, 8, 9, 12, 18 | gsumress 18641 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3939 ⟶wf 6539 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 ↾s cress 17208 +gcplusg 17232 0gc0g 17420 Σg cgsu 17421 Grpcgrp 18894 SubGrpcsubg 19079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-seq 13999 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-0g 17422 df-gsum 17423 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18897 df-subg 19082 |
This theorem is referenced by: (None) |
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