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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 19071. (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 2798 | . . 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 8577 | . . . 4 ⊢ {𝑀} ∈ Fin | |
8 | unfi 8769 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 589 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
10 | elun 4076 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
11 | gsumzunsnd.x | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
12 | elsni 4542 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
13 | gsumzunsnd.s | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
14 | 12, 13 | sylan2 595 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 = 𝑌) |
15 | gsumzunsnd.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
16 | 15 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑌 ∈ 𝐵) |
17 | 14, 16 | eqeltrd 2890 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
18 | 11, 17 | jaodan 955 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
19 | 10, 18 | sylan2b 596 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
20 | gsumzunsnd.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) | |
21 | 19, 20 | fmptd 6855 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴 ∪ {𝑀})⟶𝐵) |
22 | gsumzunsnd.c | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
23 | 11 | expcom 417 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (𝜑 → 𝑋 ∈ 𝐵)) |
24 | 15 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑌 ∈ 𝐵) |
25 | 13, 24 | eqeltrd 2890 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 ∈ 𝐵) |
26 | 25 | expcom 417 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝜑 → 𝑋 ∈ 𝐵)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑀} → (𝜑 → 𝑋 ∈ 𝐵)) |
28 | 23, 27 | jaoi 854 | . . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
29 | 10, 28 | sylbi 220 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
30 | 29 | impcom 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
31 | fvexd 6660 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
32 | 20, 9, 30, 31 | fsuppmptdm 8828 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
33 | gsumzunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
34 | disjsn 4607 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
35 | 33, 34 | sylibr 237 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
36 | eqidd 2799 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
37 | 1, 2, 3, 4, 5, 9, 21, 22, 32, 35, 36 | gsumzsplit 19040 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀})))) |
38 | 20 | reseq1i 5814 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) |
39 | ssun1 4099 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑀}) | |
40 | resmpt 5872 | . . . . . 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 2845 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
43 | 42 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) |
44 | 20 | reseq1i 5814 | . . . . 5 ⊢ (𝐹 ↾ {𝑀}) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) |
45 | ssun2 4100 | . . . . . 6 ⊢ {𝑀} ⊆ (𝐴 ∪ {𝑀}) | |
46 | resmpt 5872 | . . . . . 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 2845 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
49 | 48 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑀})) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) |
50 | 43, 49 | oveq12d 7153 | . 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 19065 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
53 | 52 | oveq2d 7151 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
54 | 37, 50, 53 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Σg cgsu 16706 Mndcmnd 17903 Cntzccntz 18437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: mplcoe5 20708 gsumzresunsn 30739 |
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