![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > gsumzresunsn | Structured version Visualization version GIF version |
Description: Append an element to a finite group sum expressed as a function restriction. (Contributed by Thierry Arnoux, 20-Nov-2023.) |
Ref | Expression |
---|---|
gsumzresunsn.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumzresunsn.p | ⊢ + = (+g‘𝐺) |
gsumzresunsn.z | ⊢ 𝑍 = (Cntz‘𝐺) |
gsumzresunsn.y | ⊢ 𝑌 = (𝐹‘𝑋) |
gsumzresunsn.f | ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) |
gsumzresunsn.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
gsumzresunsn.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
gsumzresunsn.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsumzresunsn.2 | ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐴) |
gsumzresunsn.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐶) |
gsumzresunsn.4 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
gsumzresunsn.5 | ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋})))) |
Ref | Expression |
---|---|
gsumzresunsn | ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumzresunsn.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumzresunsn.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | gsumzresunsn.z | . . 3 ⊢ 𝑍 = (Cntz‘𝐺) | |
4 | eqid 2798 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) | |
5 | gsumzresunsn.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
6 | gsumzresunsn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
7 | gsumzresunsn.5 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) ⊆ (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋})))) | |
8 | df-ima 5532 | . . . . 5 ⊢ (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝐹 ↾ (𝐴 ∪ {𝑋})) | |
9 | gsumzresunsn.f | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐶⟶𝐵) | |
10 | gsumzresunsn.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) | |
11 | gsumzresunsn.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐶) | |
12 | 11 | snssd 4702 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝐶) |
13 | 10, 12 | unssd 4113 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝑋}) ⊆ 𝐶) |
14 | 9, 13 | feqresmpt 6709 | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∪ {𝑋})) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
15 | 14 | rneqd 5772 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
16 | 8, 15 | syl5eq 2845 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
17 | 16 | fveq2d 6649 | . . . 4 ⊢ (𝜑 → (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))) = (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
18 | 7, 16, 17 | 3sstr3d 3961 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) ⊆ (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
19 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐵) |
20 | 10 | sselda 3915 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) |
21 | 19, 20 | ffvelrnd 6829 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
22 | gsumzresunsn.2 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐴) | |
23 | gsumzresunsn.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
24 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
25 | 24 | fveq2d 6649 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
26 | gsumzresunsn.y | . . . 4 ⊢ 𝑌 = (𝐹‘𝑋) | |
27 | 25, 26 | eqtr4di 2851 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = 𝑌) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19069 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
29 | 14 | oveq2d 7151 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
30 | 9, 10 | feqresmpt 6709 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
31 | 30 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))) |
32 | 31 | oveq1d 7150 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
33 | 28, 29, 32 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cun 3879 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 ran crn 5520 ↾ cres 5521 “ cima 5522 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 Basecbs 16475 +gcplusg 16557 Σg cgsu 16706 Mndcmnd 17903 Cntzccntz 18437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-seq 13365 df-hash 13687 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |