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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 14544 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 1 | oveq2d 7420 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = (𝐺 Σg {〈0, 𝑆〉})) |
3 | gsumwcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2733 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | elfvdm 6925 | . . . 4 ⊢ (𝑆 ∈ (Base‘𝐺) → 𝐺 ∈ dom Base) | |
6 | 5, 3 | eleq2s 2852 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 𝐺 ∈ dom Base) |
7 | 0nn0 12483 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
8 | nn0uz 12860 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | eleqtri 2832 | . . . 4 ⊢ 0 ∈ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 0 ∈ (ℤ≥‘0)) |
11 | 0z 12565 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
12 | f1osng 6871 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) | |
13 | 11, 12 | mpan 689 | . . . . . 6 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) |
14 | f1of 6830 | . . . . . 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 4810 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {𝑆} ⊆ 𝐵) | |
17 | 15, 16 | fssd 6732 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}⟶𝐵) |
18 | fz0sn 13597 | . . . . 5 ⊢ (0...0) = {0} | |
19 | 18 | feq2i 6706 | . . . 4 ⊢ ({〈0, 𝑆〉}:(0...0)⟶𝐵 ↔ {〈0, 𝑆〉}:{0}⟶𝐵) |
20 | 17, 19 | sylibr 233 | . . 3 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:(0...0)⟶𝐵) |
21 | 3, 4, 6, 10, 20 | gsumval2 18601 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg {〈0, 𝑆〉}) = (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0)) |
22 | fvsng 7173 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → ({〈0, 𝑆〉}‘0) = 𝑆) | |
23 | 11, 22 | mpan 689 | . . 3 ⊢ (𝑆 ∈ 𝐵 → ({〈0, 𝑆〉}‘0) = 𝑆) |
24 | 11, 23 | seq1i 13976 | . 2 ⊢ (𝑆 ∈ 𝐵 → (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0) = 𝑆) |
25 | 2, 21, 24 | 3eqtrd 2777 | 1 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {csn 4627 〈cop 4633 dom cdm 5675 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7404 0cc0 11106 ℕ0cn0 12468 ℤcz 12554 ℤ≥cuz 12818 ...cfz 13480 seqcseq 13962 〈“cs1 14541 Basecbs 17140 +gcplusg 17193 Σg cgsu 17382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-s1 14542 df-0g 17383 df-gsum 17384 |
This theorem is referenced by: gsumws2 18719 gsumccatsn 18720 gsumwspan 18723 frmdgsum 18739 frmdup2 18742 gsumwrev 19226 psgnunilem5 19355 psgnpmtr 19371 frgpup2 19637 cyc3genpmlem 32288 mrsubcv 34439 gsumws3 42881 gsumws4 42882 |
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