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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 19999. (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 2763 | . . 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 9025 | . . . 4 ⊢ {𝑀} ∈ Fin | |
| 8 | unfi 9140 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
| 9 | 6, 7, 8 | sylancl 595 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
| 10 | elun 4107 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
| 11 | gsumzunsnd.x | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
| 12 | elsni 4600 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
| 13 | gsumzunsnd.s | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
| 14 | 12, 13 | sylan2 602 | . . . . . . 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 2863 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
| 18 | 11, 17 | jaodan 970 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
| 19 | 10, 18 | sylan2b 603 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
| 20 | gsumzunsnd.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) | |
| 21 | 19, 20 | fmptd 7096 | . . 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 2863 | . . . . . . . . 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 868 | . . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
| 29 | 10, 28 | sylbi 219 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
| 30 | 29 | impcom 411 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
| 31 | fvexd 6883 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ V) | |
| 32 | 20, 9, 30, 31 | fsuppmptdm 9323 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
| 33 | gsumzunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
| 34 | disjsn 4671 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
| 35 | 33, 34 | sylibr 236 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
| 36 | eqidd 2764 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
| 37 | 1, 2, 3, 4, 5, 9, 21, 22, 32, 35, 36 | gsumzsplit 19968 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀})))) |
| 38 | 20 | reseq1i 5962 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) |
| 39 | ssun1 4131 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑀}) | |
| 40 | resmpt 6027 | . . . . . 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 | eqtrid 2810 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
| 43 | 42 | oveq2d 7413 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) |
| 44 | 20 | reseq1i 5962 | . . . . 5 ⊢ (𝐹 ↾ {𝑀}) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) |
| 45 | ssun2 4132 | . . . . . 6 ⊢ {𝑀} ⊆ (𝐴 ∪ {𝑀}) | |
| 46 | resmpt 6027 | . . . . . 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 | eqtrid 2810 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
| 49 | 48 | oveq2d 7413 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑀})) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) |
| 50 | 43, 49 | oveq12d 7415 | . 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 19993 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
| 53 | 52 | oveq2d 7413 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
| 54 | 37, 50, 53 | 3eqtrd 2802 | 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 858 = wceq 1561 ∈ wcel 2143 Vcvv 3455 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 {csn 4583 ↦ cmpt 5182 ran crn 5649 ↾ cres 5650 ‘cfv 6522 (class class class)co 7397 Fincfn 8928 Basecbs 17246 +gcplusg 17287 0gc0g 17469 Σg cgsu 17470 Mndcmnd 18769 Cntzccntz 19356 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-of 7661 df-om 7848 df-1st 7971 df-2nd 7972 df-supp 8142 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-fsupp 9309 df-oi 9459 df-card 9898 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-nn 12212 df-2 12281 df-n0 12483 df-z 12570 df-uz 12841 df-fz 13514 df-fzo 13661 df-seq 14016 df-hash 14345 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-0g 17471 df-gsum 17472 df-mre 17615 df-mrc 17616 df-acs 17618 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-submnd 18819 df-mulg 19111 df-cntz 19358 df-cmn 19823 |
| This theorem is referenced by: mplcoe5 22094 gsumzresunsn 33243 |
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