Mathbox for Thierry Arnoux |
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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 32578 | . . . 4 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
6 | 1, 5 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝐶 ∈ {𝑐 ∈ (𝐴 ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀}) |
7 | fveq1 6768 | . . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐‘𝑎) = (𝐶‘𝑎)) | |
8 | 7 | sumeq2sdv 15406 | . . . . 5 ⊢ (𝑐 = 𝐶 → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎)) |
9 | 8 | eqeq1d 2742 | . . . 4 ⊢ (𝑐 = 𝐶 → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀)) |
10 | 9 | elrab 3626 | . . 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 217 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀)) |
12 | 11 | simprd 496 | 1 ⊢ (𝜑 → Σ𝑎 ∈ (0..^𝑆)(𝐶‘𝑎) = 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 {crab 3070 ⊆ wss 3892 ‘cfv 6431 (class class class)co 7269 ↑m cmap 8590 0cc0 10864 ℕcn 11965 ℕ0cn0 12225 ℤcz 12311 ..^cfzo 13373 Σcsu 15387 reprcrepr 32576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-1st 7818 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-n0 12226 df-z 12312 df-uz 12574 df-fz 13231 df-seq 13712 df-sum 15388 df-repr 32577 |
This theorem is referenced by: reprle 32582 reprsuc 32583 reprpmtf1o 32594 hgt750lemb 32624 tgoldbachgt 32631 |
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