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Mirrors > Home > MPE Home > Th. List > gsumzunsnd | Structured version Visualization version GIF version |
Description: Append an element to a finite group sum, more general version of gsumunsnd 19078. (Contributed by AV, 7-Oct-2019.) |
Ref | Expression |
---|---|
gsumzunsnd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumzunsnd.p | ⊢ + = (+g‘𝐺) |
gsumzunsnd.z | ⊢ 𝑍 = (Cntz‘𝐺) |
gsumzunsnd.f | ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) |
gsumzunsnd.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsumzunsnd.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsumzunsnd.c | ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) |
gsumzunsnd.x | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumzunsnd.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
gsumzunsnd.d | ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) |
gsumzunsnd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumzunsnd.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) |
Ref | Expression |
---|---|
gsumzunsnd | ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzunsnd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2821 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | gsumzunsnd.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | gsumzunsnd.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
5 | gsumzunsnd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
6 | gsumzunsnd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | snfi 8594 | . . . 4 ⊢ {𝑀} ∈ Fin | |
8 | unfi 8785 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 588 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
10 | elun 4125 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
11 | gsumzunsnd.x | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
12 | elsni 4584 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
13 | gsumzunsnd.s | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
14 | 12, 13 | sylan2 594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 = 𝑌) |
15 | gsumzunsnd.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
16 | 15 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑌 ∈ 𝐵) |
17 | 14, 16 | eqeltrd 2913 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
18 | 11, 17 | jaodan 954 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
19 | 10, 18 | sylan2b 595 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
20 | gsumzunsnd.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) | |
21 | 19, 20 | fmptd 6878 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴 ∪ {𝑀})⟶𝐵) |
22 | gsumzunsnd.c | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
23 | 11 | expcom 416 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (𝜑 → 𝑋 ∈ 𝐵)) |
24 | 15 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑌 ∈ 𝐵) |
25 | 13, 24 | eqeltrd 2913 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 ∈ 𝐵) |
26 | 25 | expcom 416 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝜑 → 𝑋 ∈ 𝐵)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑀} → (𝜑 → 𝑋 ∈ 𝐵)) |
28 | 23, 27 | jaoi 853 | . . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
29 | 10, 28 | sylbi 219 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
30 | 29 | impcom 410 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
31 | fvexd 6685 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
32 | 20, 9, 30, 31 | fsuppmptdm 8844 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
33 | gsumzunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
34 | disjsn 4647 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
35 | 33, 34 | sylibr 236 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
36 | eqidd 2822 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
37 | 1, 2, 3, 4, 5, 9, 21, 22, 32, 35, 36 | gsumzsplit 19047 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀})))) |
38 | 20 | reseq1i 5849 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) |
39 | ssun1 4148 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑀}) | |
40 | resmpt 5905 | . . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∪ {𝑀}) → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
41 | 39, 40 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
42 | 38, 41 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
43 | 42 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) |
44 | 20 | reseq1i 5849 | . . . . 5 ⊢ (𝐹 ↾ {𝑀}) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) |
45 | ssun2 4149 | . . . . . 6 ⊢ {𝑀} ⊆ (𝐴 ∪ {𝑀}) | |
46 | resmpt 5905 | . . . . . 6 ⊢ ({𝑀} ⊆ (𝐴 ∪ {𝑀}) → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) | |
47 | 45, 46 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
48 | 44, 47 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
49 | 48 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑀})) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) |
50 | 43, 49 | oveq12d 7174 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀}))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
51 | gsumzunsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
52 | 1, 5, 51, 15, 13 | gsumsnd 19072 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
53 | 52 | oveq2d 7172 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
54 | 37, 50, 53 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4567 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 +gcplusg 16565 0gc0g 16713 Σg cgsu 16714 Mndcmnd 17911 Cntzccntz 18445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: mplcoe5 20249 gsumzresunsn 30691 |
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