![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > gsumsubm | 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 2193 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2193 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | gsumsubm.h | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
4 | gsumsubm.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝐺)) | |
5 | submrcl 13033 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
7 | gsumsubm.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
8 | 1 | submss 13038 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
9 | 4, 8 | syl 14 | . 2 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐺)) |
10 | gsumsubm.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
11 | eqid 2193 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 11 | subm0cl 13040 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑆) |
13 | 4, 12 | syl 14 | . 2 ⊢ (𝜑 → (0g‘𝐺) ∈ 𝑆) |
14 | 1, 2, 11 | mndlrid 13005 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐺)) → (((0g‘𝐺)(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)(0g‘𝐺)) = 𝑥)) |
15 | 6, 14 | sylan 283 | . 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 12968 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ⊆ wss 3153 ⟶wf 5242 ‘cfv 5246 (class class class)co 5910 Basecbs 12608 ↾s cress 12609 +gcplusg 12685 0gc0g 12857 Σg cgsu 12858 Mndcmnd 12987 SubMndcsubmnd 13020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-addcom 7962 ax-addass 7964 ax-i2m1 7967 ax-0lt1 7968 ax-0id 7970 ax-rnegex 7971 ax-pre-ltirr 7974 ax-pre-ltadd 7978 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-recs 6349 df-frec 6435 df-pnf 8046 df-mnf 8047 df-ltxr 8049 df-neg 8183 df-inn 8973 df-2 9031 df-z 9308 df-uz 9583 df-seqfrec 10509 df-ndx 12611 df-slot 12612 df-base 12614 df-sets 12615 df-iress 12616 df-plusg 12698 df-0g 12859 df-igsum 12860 df-mgm 12929 df-sgrp 12975 df-mnd 12988 df-submnd 13022 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |