![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 8449 | . . . 4 ⊢ {𝑀} ∈ Fin | |
6 | unfi 8638 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
7 | 4, 5, 6 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
8 | elun 4052 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
9 | gsumunsnd.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
10 | elsni 4495 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
11 | gsumunsnd.s | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
12 | 10, 11 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 = 𝑌) |
13 | gsumunsnd.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
14 | 13 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑌 ∈ 𝐵) |
15 | 12, 14 | eqeltrd 2885 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
16 | 9, 15 | jaodan 952 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
17 | 8, 16 | sylan2b 593 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
18 | gsumunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
19 | disjsn 4560 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
20 | 18, 19 | sylibr 235 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
21 | eqidd 2798 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
22 | 1, 2, 3, 7, 17, 20, 21 | gsummptfidmsplit 18774 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
23 | cmnmnd 18652 | . . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
24 | 3, 23 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
25 | gsumunsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
26 | nfv 1896 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
27 | gsumunsnfd.0 | . . . 4 ⊢ Ⅎ𝑘𝑌 | |
28 | 1, 24, 25, 13, 11, 26, 27 | gsumsnfd 18795 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
29 | 28 | oveq2d 7039 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
30 | 22, 29 | eqtrd 2833 | 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 396 ∨ wo 842 = wceq 1525 ∈ wcel 2083 Ⅎwnfc 2935 ∪ cun 3863 ∩ cin 3864 ∅c0 4217 {csn 4478 ↦ cmpt 5047 ‘cfv 6232 (class class class)co 7023 Fincfn 8364 Basecbs 16316 +gcplusg 16398 Σg cgsu 16547 Mndcmnd 17737 CMndccmn 18637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-of 7274 df-om 7444 df-1st 7552 df-2nd 7553 df-supp 7689 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fsupp 8687 df-oi 8827 df-card 9221 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-n0 11752 df-z 11836 df-uz 12098 df-fz 12747 df-fzo 12888 df-seq 13224 df-hash 13545 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-0g 16548 df-gsum 16549 df-mre 16690 df-mrc 16691 df-acs 16693 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-mulg 17986 df-cntz 18192 df-cmn 18639 |
This theorem is referenced by: gsumunsnd 18802 gsumunsnf 18803 |
Copyright terms: Public domain | W3C validator |