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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 8522 | . . . . 5 ⊢ ( ∼ Er 𝐴 → (𝐴 ∈ Fin → ∼ ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
5 | 1, 4 | uniqs2 8568 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / ∼ ) = 𝐴) |
6 | 5 | fveq2d 6778 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = (♯‘𝐴)) |
7 | pwfi 8961 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
8 | 2, 7 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
9 | 1 | qsss 8567 | . . . 4 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ 𝒫 𝐴) |
10 | 8, 9 | ssfid 9042 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ∈ Fin) |
11 | elpwi 4542 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
12 | ssfi 8956 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
13 | 12 | ex 413 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝑥 ⊆ 𝐴 → 𝑥 ∈ Fin)) |
14 | 2, 11, 13 | syl2im 40 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ Fin)) |
15 | 14 | ssrdv 3927 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ⊆ Fin) |
16 | 9, 15 | sstrd 3931 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ Fin) |
17 | qsdisj2 8584 | . . . 4 ⊢ ( ∼ Er 𝐴 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) |
19 | 10, 16, 18 | hashuni 15538 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
20 | 6, 19 | eqtr3d 2780 | 1 ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⊆ wss 3887 𝒫 cpw 4533 ∪ cuni 4839 Disj wdisj 5039 ‘cfv 6433 Er wer 8495 / cqs 8497 Fincfn 8733 ♯chash 14044 Σcsu 15397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 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 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 |
This theorem is referenced by: lagsubg2 18817 sylow1lem3 19205 sylow2a 19224 hashclwwlkn0 28438 |
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