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| Mirrors > Home > MPE Home > Th. List > Mathboxes > reprsum | Structured version Visualization version GIF version | ||
| Description: Sums of values of the members of the representation of 𝑀 equal 𝑀. (Contributed by Thierry Arnoux, 5-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| reprval.a | ⊢ (𝜑 → 𝐴 ⊆ ℕ) | 
| reprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| reprval.s | ⊢ (𝜑 → 𝑆 ∈ ℕ0) | 
| reprf.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) | 
| Ref | Expression | 
|---|---|
| reprsum | ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reprf.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴(repr‘𝑆)𝑀)) | |
| 2 | reprval.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ) | |
| 3 | reprval.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 4 | reprval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℕ0) | |
| 5 | 2, 3, 4 | reprval 34626 | . . . 4 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) | 
| 6 | 1, 5 | eleqtrd 2842 | . . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) | 
| 7 | fveq1 6904 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑎) = (𝐶‘𝑎)) | |
| 8 | 7 | sumeq2sdv 15740 | . . . . 5 ⊢ (𝑐 = 𝐶 → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎)) | 
| 9 | 8 | eqeq1d 2738 | . . . 4 ⊢ (𝑐 = 𝐶 → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀)) | 
| 10 | 9 | elrab 3691 | . . 3 ⊢ (𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ↔ (𝐶 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀)) | 
| 11 | 6, 10 | sylib 218 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀)) | 
| 12 | 11 | simprd 495 | 1 ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 0cc0 11156 ℕcn 12267 ℕ0cn0 12528 ℤcz 12615 ..^cfzo 13695 Σcsu 15723 reprcrepr 34624 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-addcl 11216 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-neg 11496 df-nn 12268 df-z 12616 df-seq 14044 df-sum 15724 df-repr 34625 | 
| This theorem is referenced by: reprle 34630 reprsuc 34631 reprpmtf1o 34642 hgt750lemb 34672 tgoldbachgt 34679 | 
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