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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 13947 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 1 | oveq2d 7165 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = (𝐺 Σg {〈0, 𝑆〉})) |
3 | gsumwcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2820 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | elfvdm 6695 | . . . 4 ⊢ (𝑆 ∈ (Base‘𝐺) → 𝐺 ∈ dom Base) | |
6 | 5, 3 | eleq2s 2930 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 𝐺 ∈ dom Base) |
7 | 0nn0 11906 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
8 | nn0uz 12274 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | eleqtri 2910 | . . . 4 ⊢ 0 ∈ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 0 ∈ (ℤ≥‘0)) |
11 | 0z 11986 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
12 | f1osng 6648 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) | |
13 | 11, 12 | mpan 688 | . . . . . 6 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) |
14 | f1of 6608 | . . . . . 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 4734 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {𝑆} ⊆ 𝐵) | |
17 | 15, 16 | fssd 6521 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}⟶𝐵) |
18 | fz0sn 13004 | . . . . 5 ⊢ (0...0) = {0} | |
19 | 18 | feq2i 6499 | . . . 4 ⊢ ({〈0, 𝑆〉}:(0...0)⟶𝐵 ↔ {〈0, 𝑆〉}:{0}⟶𝐵) |
20 | 17, 19 | sylibr 236 | . . 3 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:(0...0)⟶𝐵) |
21 | 3, 4, 6, 10, 20 | gsumval2 17891 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg {〈0, 𝑆〉}) = (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0)) |
22 | fvsng 6935 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → ({〈0, 𝑆〉}‘0) = 𝑆) | |
23 | 11, 22 | mpan 688 | . . 3 ⊢ (𝑆 ∈ 𝐵 → ({〈0, 𝑆〉}‘0) = 𝑆) |
24 | 11, 23 | seq1i 13380 | . 2 ⊢ (𝑆 ∈ 𝐵 → (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0) = 𝑆) |
25 | 2, 21, 24 | 3eqtrd 2859 | 1 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {csn 4560 〈cop 4566 dom cdm 5548 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7149 0cc0 10530 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12889 seqcseq 13366 〈“cs1 13944 Basecbs 16478 +gcplusg 16560 Σg cgsu 16709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-seq 13367 df-s1 13945 df-0g 16710 df-gsum 16711 |
This theorem is referenced by: gsumws2 18002 gsumccatsn 18003 gsumwspan 18006 frmdgsum 18022 frmdup2 18025 gsumwrev 18489 psgnunilem5 18617 psgnpmtr 18633 frgpup2 18897 cyc3genpmlem 30814 mrsubcv 32778 gsumws3 40623 gsumws4 40624 |
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