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Mirrors > Home > MPE Home > Th. List > gsumws1 | Structured version Visualization version GIF version |
Description: A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
gsumwcl.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
gsumws1 | ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 13796 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 1 | oveq2d 7032 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = (𝐺 Σg {〈0, 𝑆〉})) |
3 | gsumwcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2795 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | elfvdm 6570 | . . . 4 ⊢ (𝑆 ∈ (Base‘𝐺) → 𝐺 ∈ dom Base) | |
6 | 5, 3 | eleq2s 2901 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 𝐺 ∈ dom Base) |
7 | 0nn0 11760 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
8 | nn0uz 12129 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | eleqtri 2881 | . . . 4 ⊢ 0 ∈ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 0 ∈ (ℤ≥‘0)) |
11 | 0z 11840 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
12 | f1osng 6523 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) | |
13 | 11, 12 | mpan 686 | . . . . . 6 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) |
14 | f1of 6483 | . . . . . 6 ⊢ ({〈0, 𝑆〉}:{0}–1-1-onto→{𝑆} → {〈0, 𝑆〉}:{0}⟶{𝑆}) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}⟶{𝑆}) |
16 | snssi 4648 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {𝑆} ⊆ 𝐵) | |
17 | 15, 16 | fssd 6396 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}⟶𝐵) |
18 | fz0sn 12857 | . . . . 5 ⊢ (0...0) = {0} | |
19 | 18 | feq2i 6374 | . . . 4 ⊢ ({〈0, 𝑆〉}:(0...0)⟶𝐵 ↔ {〈0, 𝑆〉}:{0}⟶𝐵) |
20 | 17, 19 | sylibr 235 | . . 3 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:(0...0)⟶𝐵) |
21 | 3, 4, 6, 10, 20 | gsumval2 17719 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg {〈0, 𝑆〉}) = (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0)) |
22 | fvsng 6805 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → ({〈0, 𝑆〉}‘0) = 𝑆) | |
23 | 11, 22 | mpan 686 | . . 3 ⊢ (𝑆 ∈ 𝐵 → ({〈0, 𝑆〉}‘0) = 𝑆) |
24 | 11, 23 | seq1i 13233 | . 2 ⊢ (𝑆 ∈ 𝐵 → (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0) = 𝑆) |
25 | 2, 21, 24 | 3eqtrd 2835 | 1 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 {csn 4472 〈cop 4478 dom cdm 5443 ⟶wf 6221 –1-1-onto→wf1o 6224 ‘cfv 6225 (class class class)co 7016 0cc0 10383 ℕ0cn0 11745 ℤcz 11829 ℤ≥cuz 12093 ...cfz 12742 seqcseq 13219 〈“cs1 13793 Basecbs 16312 +gcplusg 16394 Σg cgsu 16543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-seq 13220 df-s1 13794 df-0g 16544 df-gsum 16545 |
This theorem is referenced by: gsumws2 17818 gsumccatsn 17819 gsumwspan 17822 frmdgsum 17838 frmdup2 17841 gsumwrev 18235 psgnunilem5 18353 psgnpmtr 18369 frgpup2 18629 cyc3genpmlem 30431 mrsubcv 32365 gsumws3 40035 gsumws4 40036 |
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