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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 8753 | . . . . 5 ⊢ ( ∼ Er 𝐴 → (𝐴 ∈ Fin → ∼ ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
5 | 1, 4 | uniqs2 8802 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / ∼ ) = 𝐴) |
6 | 5 | fveq2d 6904 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = (♯‘𝐴)) |
7 | pwfi 9207 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
8 | 2, 7 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
9 | 1 | qsss 8801 | . . . 4 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ 𝒫 𝐴) |
10 | 8, 9 | ssfid 9296 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ∈ Fin) |
11 | elpwi 4611 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
12 | ssfi 9202 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
13 | 12 | ex 411 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝑥 ⊆ 𝐴 → 𝑥 ∈ Fin)) |
14 | 2, 11, 13 | syl2im 40 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ Fin)) |
15 | 14 | ssrdv 3986 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ⊆ Fin) |
16 | 9, 15 | sstrd 3990 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ Fin) |
17 | qsdisj2 8818 | . . . 4 ⊢ ( ∼ Er 𝐴 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) |
19 | 10, 16, 18 | hashuni 15810 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
20 | 6, 19 | eqtr3d 2769 | 1 ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⊆ wss 3947 𝒫 cpw 4604 ∪ cuni 4910 Disj wdisj 5115 ‘cfv 6551 Er wer 8726 / cqs 8728 Fincfn 8968 ♯chash 14327 Σcsu 15670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-disj 5116 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-er 8729 df-ec 8731 df-qs 8735 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-sup 9471 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 |
This theorem is referenced by: lagsubg2 19154 sylow1lem3 19560 sylow2a 19579 hashclwwlkn0 29902 |
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