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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sumsnd | Structured version Visualization version GIF version | ||
| Description: A sum of a singleton is the term. The deduction version of sumsn 15782. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
| Ref | Expression |
|---|---|
| sumsnd.1 | ⊢ (𝜑 → Ⅎ𝑘𝐵) |
| sumsnd.2 | ⊢ Ⅎ𝑘𝜑 |
| sumsnd.3 | ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵) |
| sumsnd.4 | ⊢ (𝜑 → 𝑀 ∈ 𝑉) |
| sumsnd.5 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| Ref | Expression |
|---|---|
| sumsnd | ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | csbeq1a 3913 | . . . 4 ⊢ (𝑘 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑘⦌𝐴) | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑚𝐴 | |
| 3 | nfcsb1v 3923 | . . . 4 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐴 | |
| 4 | 1, 2, 3 | cbvsum 15731 | . . 3 ⊢ Σ𝑘 ∈ {𝑀}𝐴 = Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 |
| 5 | csbeq1 3902 | . . . 4 ⊢ (𝑚 = ({〈1, 𝑀〉}‘𝑛) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) | |
| 6 | 1nn 12277 | . . . . 5 ⊢ 1 ∈ ℕ | |
| 7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 1 ∈ ℕ) |
| 8 | sumsnd.4 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑉) | |
| 9 | f1osng 6889 | . . . . . 6 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) | |
| 10 | 6, 8, 9 | sylancr 587 | . . . . 5 ⊢ (𝜑 → {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) |
| 11 | 1z 12647 | . . . . . 6 ⊢ 1 ∈ ℤ | |
| 12 | fzsn 13606 | . . . . . 6 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
| 13 | f1oeq2 6837 | . . . . . 6 ⊢ ((1...1) = {1} → ({〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀} ↔ {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀})) | |
| 14 | 11, 12, 13 | mp2b 10 | . . . . 5 ⊢ ({〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀} ↔ {〈1, 𝑀〉}:{1}–1-1-onto→{𝑀}) |
| 15 | 10, 14 | sylibr 234 | . . . 4 ⊢ (𝜑 → {〈1, 𝑀〉}:(1...1)–1-1-onto→{𝑀}) |
| 16 | elsni 4643 | . . . . . . 7 ⊢ (𝑚 ∈ {𝑀} → 𝑚 = 𝑀) | |
| 17 | 16 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝑚 = 𝑀) |
| 18 | 17 | csbeq1d 3903 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
| 19 | sumsnd.2 | . . . . . . . 8 ⊢ Ⅎ𝑘𝜑 | |
| 20 | sumsnd.1 | . . . . . . . 8 ⊢ (𝜑 → Ⅎ𝑘𝐵) | |
| 21 | sumsnd.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵) | |
| 22 | 19, 20, 8, 21 | csbiedf 3929 | . . . . . . 7 ⊢ (𝜑 → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
| 23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
| 24 | sumsnd.5 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 25 | 24 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → 𝐵 ∈ ℂ) |
| 26 | 23, 25 | eqeltrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑀 / 𝑘⦌𝐴 ∈ ℂ) |
| 27 | 18, 26 | eqeltrd 2841 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ {𝑀}) → ⦋𝑚 / 𝑘⦌𝐴 ∈ ℂ) |
| 28 | 22 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋𝑀 / 𝑘⦌𝐴 = 𝐵) |
| 29 | elfz1eq 13575 | . . . . . . . 8 ⊢ (𝑛 ∈ (1...1) → 𝑛 = 1) | |
| 30 | 29 | fveq2d 6910 | . . . . . . 7 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝑀〉}‘𝑛) = ({〈1, 𝑀〉}‘1)) |
| 31 | fvsng 7200 | . . . . . . . 8 ⊢ ((1 ∈ ℕ ∧ 𝑀 ∈ 𝑉) → ({〈1, 𝑀〉}‘1) = 𝑀) | |
| 32 | 6, 8, 31 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → ({〈1, 𝑀〉}‘1) = 𝑀) |
| 33 | 30, 32 | sylan9eqr 2799 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝑀〉}‘𝑛) = 𝑀) |
| 34 | 33 | csbeq1d 3903 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴 = ⦋𝑀 / 𝑘⦌𝐴) |
| 35 | 29 | fveq2d 6910 | . . . . . 6 ⊢ (𝑛 ∈ (1...1) → ({〈1, 𝐵〉}‘𝑛) = ({〈1, 𝐵〉}‘1)) |
| 36 | fvsng 7200 | . . . . . . 7 ⊢ ((1 ∈ ℕ ∧ 𝐵 ∈ ℂ) → ({〈1, 𝐵〉}‘1) = 𝐵) | |
| 37 | 6, 24, 36 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → ({〈1, 𝐵〉}‘1) = 𝐵) |
| 38 | 35, 37 | sylan9eqr 2799 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = 𝐵) |
| 39 | 28, 34, 38 | 3eqtr4rd 2788 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...1)) → ({〈1, 𝐵〉}‘𝑛) = ⦋({〈1, 𝑀〉}‘𝑛) / 𝑘⦌𝐴) |
| 40 | 5, 7, 15, 27, 39 | fsum 15756 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ {𝑀}⦋𝑚 / 𝑘⦌𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
| 41 | 4, 40 | eqtrid 2789 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = (seq1( + , {〈1, 𝐵〉})‘1)) |
| 42 | 11, 37 | seq1i 14056 | . 2 ⊢ (𝜑 → (seq1( + , {〈1, 𝐵〉})‘1) = 𝐵) |
| 43 | 41, 42 | eqtrd 2777 | 1 ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ⦋csb 3899 {csn 4626 〈cop 4632 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 ℕcn 12266 ℤcz 12613 ...cfz 13547 seqcseq 14042 Σcsu 15722 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 |
| This theorem is referenced by: sumpair 45040 dvnmul 45958 sge0sn 46394 hoidmvlelem3 46612 |
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