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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 14551 | . . 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 2726 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | elfvdm 6921 | . . . 4 ⊢ (𝑆 ∈ (Base‘𝐺) → 𝐺 ∈ dom Base) | |
6 | 5, 3 | eleq2s 2845 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 𝐺 ∈ dom Base) |
7 | 0nn0 12488 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
8 | nn0uz 12865 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | eleqtri 2825 | . . . 4 ⊢ 0 ∈ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 0 ∈ (ℤ≥‘0)) |
11 | 0z 12570 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
12 | f1osng 6867 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → {⟨0, 𝑆⟩}:{0}–1-1-onto→{𝑆}) | |
13 | 11, 12 | mpan 687 | . . . . . 6 ⊢ (𝑆 ∈ 𝐵 → {⟨0, 𝑆⟩}:{0}–1-1-onto→{𝑆}) |
14 | f1of 6826 | . . . . . 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 4806 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {𝑆} ⊆ 𝐵) | |
17 | 15, 16 | fssd 6728 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → {⟨0, 𝑆⟩}:{0}⟶𝐵) |
18 | fz0sn 13604 | . . . . 5 ⊢ (0...0) = {0} | |
19 | 18 | feq2i 6702 | . . . 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 18616 | . 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 687 | . . 3 ⊢ (𝑆 ∈ 𝐵 → ({⟨0, 𝑆⟩}‘0) = 𝑆) |
24 | 11, 23 | seq1i 13983 | . 2 ⊢ (𝑆 ∈ 𝐵 → (seq0((+g‘𝐺), {⟨0, 𝑆⟩})‘0) = 𝑆) |
25 | 2, 21, 24 | 3eqtrd 2770 | 1 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg ⟨“𝑆”⟩) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4623 ⟨cop 4629 dom cdm 5669 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 0cc0 11109 ℕ0cn0 12473 ℤcz 12559 ℤ≥cuz 12823 ...cfz 13487 seqcseq 13969 ⟨“cs1 14548 Basecbs 17150 +gcplusg 17203 Σg cgsu 17392 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-seq 13970 df-s1 14549 df-0g 17393 df-gsum 17394 |
This theorem is referenced by: gsumws2 18764 gsumccatsn 18765 gsumwspan 18768 frmdgsum 18784 frmdup2 18787 gsumwrev 19282 psgnunilem5 19411 psgnpmtr 19427 frgpup2 19693 cyc3genpmlem 32813 mrsubcv 35028 gsumws3 43506 gsumws4 43507 |
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