Mathbox for Thierry Arnoux |
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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 2821 | . . 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 5568 | . . . . 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 4742 | . . . . . . . 8 ⊢ (𝜑 → {𝑋} ⊆ 𝐶) |
13 | 10, 12 | unssd 4162 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∪ {𝑋}) ⊆ 𝐶) |
14 | 9, 13 | feqresmpt 6734 | . . . . . 6 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∪ {𝑋})) = (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
15 | 14 | rneqd 5808 | . . . . 5 ⊢ (𝜑 → ran (𝐹 ↾ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
16 | 8, 15 | syl5eq 2868 | . . . 4 ⊢ (𝜑 → (𝐹 “ (𝐴 ∪ {𝑋})) = ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) |
17 | 16 | fveq2d 6674 | . . . 4 ⊢ (𝜑 → (𝑍‘(𝐹 “ (𝐴 ∪ {𝑋}))) = (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
18 | 7, 16, 17 | 3sstr3d 4013 | . . 3 ⊢ (𝜑 → ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)) ⊆ (𝑍‘ran (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
19 | 9 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹:𝐶⟶𝐵) |
20 | 10 | sselda 3967 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐶) |
21 | 19, 20 | ffvelrnd 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵) |
22 | gsumzresunsn.2 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐴) | |
23 | gsumzresunsn.4 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
24 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋) | |
25 | 24 | fveq2d 6674 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋)) |
26 | gsumzresunsn.y | . . . 4 ⊢ 𝑌 = (𝐹‘𝑋) | |
27 | 25, 26 | syl6eqr 2874 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = 𝑌) |
28 | 1, 2, 3, 4, 5, 6, 18, 21, 11, 22, 23, 27 | gsumzunsnd 19076 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥))) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
29 | 14 | oveq2d 7172 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ∪ {𝑋}))) = (𝐺 Σg (𝑥 ∈ (𝐴 ∪ {𝑋}) ↦ (𝐹‘𝑥)))) |
30 | 9, 10 | feqresmpt 6734 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
31 | 30 | oveq2d 7172 | . . 3 ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐴)) = (𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))) |
32 | 31 | oveq1d 7171 | . 2 ⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐴)) + 𝑌) = ((𝐺 Σg (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) + 𝑌)) |
33 | 28, 29, 32 | 3eqtr4d 2866 | 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 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ⊆ wss 3936 {csn 4567 ↦ cmpt 5146 ran crn 5556 ↾ cres 5557 “ cima 5558 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Fincfn 8509 Basecbs 16483 +gcplusg 16565 Σg cgsu 16714 Mndcmnd 17911 Cntzccntz 18445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-gsum 16716 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 |
This theorem is referenced by: (None) |
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