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Mirrors > Home > MPE Home > Th. List > gsumunsnfd | Structured version Visualization version GIF version |
Description: Append an element to a finite group sum, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 11-Dec-2019.) |
Ref | Expression |
---|---|
gsumunsnd.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumunsnd.p | ⊢ + = (+g‘𝐺) |
gsumunsnd.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumunsnd.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsumunsnd.f | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumunsnd.m | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
gsumunsnd.d | ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) |
gsumunsnd.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumunsnd.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) |
gsumunsnfd.0 | ⊢ Ⅎ𝑘𝑌 |
Ref | Expression |
---|---|
gsumunsnfd | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumunsnd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumunsnd.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | gsumunsnd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsumunsnd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | snfi 8994 | . . . 4 ⊢ {𝑀} ∈ Fin | |
6 | unfi 9122 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
7 | 4, 5, 6 | sylancl 587 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
8 | elun 4112 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
9 | gsumunsnd.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
10 | elsni 4607 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
11 | gsumunsnd.s | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
12 | 10, 11 | sylan2 594 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 = 𝑌) |
13 | gsumunsnd.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 13 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑌 ∈ 𝐵) |
15 | 12, 14 | eqeltrd 2834 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
16 | 9, 15 | jaodan 957 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
17 | 8, 16 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
18 | gsumunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
19 | disjsn 4676 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
20 | 18, 19 | sylibr 233 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
21 | eqidd 2734 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
22 | 1, 2, 3, 7, 17, 20, 21 | gsummptfidmsplit 19715 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
23 | cmnmnd 19587 | . . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
25 | gsumunsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
26 | nfv 1918 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
27 | gsumunsnfd.0 | . . . 4 ⊢ Ⅎ𝑘𝑌 | |
28 | 1, 24, 25, 13, 11, 26, 27 | gsumsnfd 19736 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
29 | 28 | oveq2d 7377 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
30 | 22, 29 | eqtrd 2773 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 ∪ cun 3912 ∩ cin 3913 ∅c0 4286 {csn 4590 ↦ cmpt 5192 ‘cfv 6500 (class class class)co 7361 Fincfn 8889 Basecbs 17091 +gcplusg 17141 Σg cgsu 17330 Mndcmnd 18564 CMndccmn 19570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-seq 13916 df-hash 14240 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-0g 17331 df-gsum 17332 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 |
This theorem is referenced by: gsumunsnd 19743 gsumunsnf 19744 |
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