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Mirrors > Home > MPE Home > Th. List > qshash | Structured version Visualization version GIF version |
Description: The cardinality of a set with an equivalence relation is the sum of the cardinalities of its equivalence classes. (Contributed by Mario Carneiro, 16-Jan-2015.) |
Ref | Expression |
---|---|
qshash.1 | ⊢ (𝜑 → ∼ Er 𝐴) |
qshash.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
Ref | Expression |
---|---|
qshash | ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qshash.1 | . . . 4 ⊢ (𝜑 → ∼ Er 𝐴) | |
2 | qshash.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
3 | erex 8296 | . . . . 5 ⊢ ( ∼ Er 𝐴 → (𝐴 ∈ Fin → ∼ ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
5 | 1, 4 | uniqs2 8342 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / ∼ ) = 𝐴) |
6 | 5 | fveq2d 6649 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = (♯‘𝐴)) |
7 | pwfi 8803 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
8 | 2, 7 | sylib 221 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
9 | 1 | qsss 8341 | . . . 4 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ 𝒫 𝐴) |
10 | 8, 9 | ssfid 8725 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ∈ Fin) |
11 | elpwi 4506 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
12 | ssfi 8722 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
13 | 12 | ex 416 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝑥 ⊆ 𝐴 → 𝑥 ∈ Fin)) |
14 | 2, 11, 13 | syl2im 40 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ Fin)) |
15 | 14 | ssrdv 3921 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ⊆ Fin) |
16 | 9, 15 | sstrd 3925 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ Fin) |
17 | qsdisj2 8358 | . . . 4 ⊢ ( ∼ Er 𝐴 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) |
19 | 10, 16, 18 | hashuni 15173 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
20 | 6, 19 | eqtr3d 2835 | 1 ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 𝒫 cpw 4497 ∪ cuni 4800 Disj wdisj 4995 ‘cfv 6324 Er wer 8269 / cqs 8271 Fincfn 8492 ♯chash 13686 Σcsu 15034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 |
This theorem is referenced by: lagsubg2 18333 sylow1lem3 18717 sylow2a 18736 hashclwwlkn0 27859 |
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