Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumdifsndf | Structured version Visualization version GIF version |
Description: Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.) |
Ref | Expression |
---|---|
gsumdifsndf.k | ⊢ Ⅎ𝑘𝑌 |
gsumdifsndf.n | ⊢ Ⅎ𝑘𝜑 |
gsumdifsndf.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumdifsndf.p | ⊢ + = (+g‘𝐺) |
gsumdifsndf.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsumdifsndf.a | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
gsumdifsndf.f | ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) |
gsumdifsndf.e | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) |
gsumdifsndf.m | ⊢ (𝜑 → 𝑀 ∈ 𝐴) |
gsumdifsndf.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumdifsndf.s | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) |
Ref | Expression |
---|---|
gsumdifsndf | ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumdifsndf.n | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | gsumdifsndf.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2610 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
4 | gsumdifsndf.p | . . 3 ⊢ + = (+g‘𝐺) | |
5 | gsumdifsndf.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
6 | gsumdifsndf.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
7 | gsumdifsndf.e | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵) | |
8 | gsumdifsndf.f | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺)) | |
9 | difid 3902 | . . . 4 ⊢ ({𝑀} ∖ {𝑀}) = ∅ | |
10 | gsumdifsndf.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝐴) | |
11 | 10 | snssd 4281 | . . . . 5 ⊢ (𝜑 → {𝑀} ⊆ 𝐴) |
12 | difin2 3849 | . . . . 5 ⊢ ({𝑀} ⊆ 𝐴 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝜑 → ({𝑀} ∖ {𝑀}) = ((𝐴 ∖ {𝑀}) ∩ {𝑀})) |
14 | 9, 13 | syl5reqr 2659 | . . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∩ {𝑀}) = ∅) |
15 | difsnid 4282 | . . . . 5 ⊢ (𝑀 ∈ 𝐴 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) | |
16 | 10, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∖ {𝑀}) ∪ {𝑀}) = 𝐴) |
17 | 16 | eqcomd 2616 | . . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∖ {𝑀}) ∪ {𝑀})) |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 14, 17 | gsumsplit2f 41936 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)))) |
19 | cmnmnd 18031 | . . . . 5 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd) | |
20 | 5, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd) |
21 | gsumdifsndf.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
22 | gsumdifsndf.s | . . . 4 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌) | |
23 | gsumdifsndf.k | . . . 4 ⊢ Ⅎ𝑘𝑌 | |
24 | 2, 20, 10, 21, 22, 1, 23 | gsumsnfd 18174 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋)) = 𝑌) |
25 | 24 | oveq2d 6565 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ {𝑀} ↦ 𝑋))) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
26 | 18, 25 | eqtrd 2644 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 Ⅎwnfc 2738 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 finSupp cfsupp 8158 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Σg cgsu 15924 Mndcmnd 17117 CMndccmn 18016 |
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: (None) |
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