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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 13231 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 1 | oveq2d 6565 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = (𝐺 Σg {〈0, 𝑆〉})) |
3 | gsumwcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
5 | elfvdm 6130 | . . . 4 ⊢ (𝑆 ∈ (Base‘𝐺) → 𝐺 ∈ dom Base) | |
6 | 5, 3 | eleq2s 2706 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 𝐺 ∈ dom Base) |
7 | 0nn0 11184 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
8 | nn0uz 11598 | . . . . 5 ⊢ ℕ0 = (ℤ≥‘0) | |
9 | 7, 8 | eleqtri 2686 | . . . 4 ⊢ 0 ∈ (ℤ≥‘0) |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑆 ∈ 𝐵 → 0 ∈ (ℤ≥‘0)) |
11 | 0z 11265 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
12 | f1osng 6089 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) | |
13 | 11, 12 | mpan 702 | . . . . . 6 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}–1-1-onto→{𝑆}) |
14 | f1of 6050 | . . . . . 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 4280 | . . . . 5 ⊢ (𝑆 ∈ 𝐵 → {𝑆} ⊆ 𝐵) | |
17 | 15, 16 | fssd 5970 | . . . 4 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:{0}⟶𝐵) |
18 | fzsn 12254 | . . . . . 6 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
19 | 11, 18 | ax-mp 5 | . . . . 5 ⊢ (0...0) = {0} |
20 | 19 | feq2i 5950 | . . . 4 ⊢ ({〈0, 𝑆〉}:(0...0)⟶𝐵 ↔ {〈0, 𝑆〉}:{0}⟶𝐵) |
21 | 17, 20 | sylibr 223 | . . 3 ⊢ (𝑆 ∈ 𝐵 → {〈0, 𝑆〉}:(0...0)⟶𝐵) |
22 | 3, 4, 6, 10, 21 | gsumval2 17103 | . 2 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg {〈0, 𝑆〉}) = (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0)) |
23 | fvsng 6352 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐵) → ({〈0, 𝑆〉}‘0) = 𝑆) | |
24 | 11, 23 | mpan 702 | . . 3 ⊢ (𝑆 ∈ 𝐵 → ({〈0, 𝑆〉}‘0) = 𝑆) |
25 | 11, 24 | seq1i 12677 | . 2 ⊢ (𝑆 ∈ 𝐵 → (seq0((+g‘𝐺), {〈0, 𝑆〉})‘0) = 𝑆) |
26 | 2, 22, 25 | 3eqtrd 2648 | 1 ⊢ (𝑆 ∈ 𝐵 → (𝐺 Σg 〈“𝑆”〉) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 〈cop 4131 dom cdm 5038 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 〈“cs1 13149 Basecbs 15695 +gcplusg 15768 Σg cgsu 15924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-seq 12664 df-s1 13157 df-0g 15925 df-gsum 15926 |
This theorem is referenced by: gsumws2 17202 gsumccatsn 17203 gsumwspan 17206 frmdgsum 17222 frmdup2 17225 gsumwrev 17619 psgnunilem5 17737 psgnpmtr 17753 frgpup2 18012 mrsubcv 30661 gsumws3 37521 gsumws4 37522 |
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