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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 18180. (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 2610 | . . 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 7923 | . . . 4 ⊢ {𝑀} ∈ Fin | |
8 | unfi 8112 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑀} ∈ Fin) → (𝐴 ∪ {𝑀}) ∈ Fin) | |
9 | 6, 7, 8 | sylancl 693 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) ∈ Fin) |
10 | elun 3715 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) ↔ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) | |
11 | gsumzunsnd.x | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
12 | elsni 4142 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) | |
13 | gsumzunsnd.s | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
14 | 12, 13 | sylan2 490 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 = 𝑌) |
15 | gsumzunsnd.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
16 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑌 ∈ 𝐵) |
17 | 14, 16 | eqeltrd 2688 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝑋 ∈ 𝐵) |
18 | 11, 17 | jaodan 822 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀})) → 𝑋 ∈ 𝐵) |
19 | 10, 18 | sylan2b 491 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
20 | gsumzunsnd.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) | |
21 | 19, 20 | fmptd 6292 | . . 3 ⊢ (𝜑 → 𝐹:(𝐴 ∪ {𝑀})⟶𝐵) |
22 | gsumzunsnd.c | . . 3 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹)) | |
23 | 11 | expcom 450 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → (𝜑 → 𝑋 ∈ 𝐵)) |
24 | 15 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑌 ∈ 𝐵) |
25 | 13, 24 | eqeltrd 2688 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 ∈ 𝐵) |
26 | 25 | expcom 450 | . . . . . . . 8 ⊢ (𝑘 = 𝑀 → (𝜑 → 𝑋 ∈ 𝐵)) |
27 | 12, 26 | syl 17 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑀} → (𝜑 → 𝑋 ∈ 𝐵)) |
28 | 23, 27 | jaoi 393 | . . . . . 6 ⊢ ((𝑘 ∈ 𝐴 ∨ 𝑘 ∈ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
29 | 10, 28 | sylbi 206 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∪ {𝑀}) → (𝜑 → 𝑋 ∈ 𝐵)) |
30 | 29 | impcom 445 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∪ {𝑀})) → 𝑋 ∈ 𝐵) |
31 | fvex 6113 | . . . . 5 ⊢ (0g‘𝐺) ∈ V | |
32 | 31 | a1i 11 | . . . 4 ⊢ (𝜑 → (0g‘𝐺) ∈ V) |
33 | 20, 9, 30, 32 | fsuppmptdm 8169 | . . 3 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝐺)) |
34 | gsumzunsnd.d | . . . 4 ⊢ (𝜑 → ¬ 𝑀 ∈ 𝐴) | |
35 | disjsn 4192 | . . . 4 ⊢ ((𝐴 ∩ {𝑀}) = ∅ ↔ ¬ 𝑀 ∈ 𝐴) | |
36 | 34, 35 | sylibr 223 | . . 3 ⊢ (𝜑 → (𝐴 ∩ {𝑀}) = ∅) |
37 | eqidd 2611 | . . 3 ⊢ (𝜑 → (𝐴 ∪ {𝑀}) = (𝐴 ∪ {𝑀})) | |
38 | 1, 2, 3, 4, 5, 9, 21, 22, 33, 36, 37 | gsumzsplit 18150 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀})))) |
39 | 20 | reseq1i 5313 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) |
40 | ssun1 3738 | . . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ {𝑀}) | |
41 | resmpt 5369 | . . . . . 6 ⊢ (𝐴 ⊆ (𝐴 ∪ {𝑀}) → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) | |
42 | 40, 41 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
43 | 39, 42 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝑋)) |
44 | 43 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋))) |
45 | 20 | reseq1i 5313 | . . . . 5 ⊢ (𝐹 ↾ {𝑀}) = ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) |
46 | ssun2 3739 | . . . . . 6 ⊢ {𝑀} ⊆ (𝐴 ∪ {𝑀}) | |
47 | resmpt 5369 | . . . . . 6 ⊢ ({𝑀} ⊆ (𝐴 ∪ {𝑀}) → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) | |
48 | 46, 47 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ (𝐴 ∪ {𝑀}) ↦ 𝑋) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
49 | 45, 48 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝑋)) |
50 | 49 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ {𝑀})) = (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) |
51 | 44, 50 | oveq12d 6567 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐴)) + (𝐺 Σg (𝐹 ↾ {𝑀}))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
52 | gsumzunsnd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
53 | 1, 5, 52, 15, 13 | gsumsnd 18175 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
54 | 53 | oveq2d 6565 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
55 | 38, 51, 54 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 ↦ cmpt 4643 ran crn 5039 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 Cntzccntz 17571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 |
This theorem is referenced by: mplcoe5 19289 |
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