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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 8726 | . . . . 5 ⊢ ( ∼ Er 𝐴 → (𝐴 ∈ Fin → ∼ ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
5 | 1, 4 | uniqs2 8772 | . . 3 ⊢ (𝜑 → ∪ (𝐴 / ∼ ) = 𝐴) |
6 | 5 | fveq2d 6888 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = (♯‘𝐴)) |
7 | pwfi 9177 | . . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) | |
8 | 2, 7 | sylib 217 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ∈ Fin) |
9 | 1 | qsss 8771 | . . . 4 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ 𝒫 𝐴) |
10 | 8, 9 | ssfid 9266 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ∈ Fin) |
11 | elpwi 4604 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
12 | ssfi 9172 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ⊆ 𝐴) → 𝑥 ∈ Fin) | |
13 | 12 | ex 412 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (𝑥 ⊆ 𝐴 → 𝑥 ∈ Fin)) |
14 | 2, 11, 13 | syl2im 40 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ Fin)) |
15 | 14 | ssrdv 3983 | . . . 4 ⊢ (𝜑 → 𝒫 𝐴 ⊆ Fin) |
16 | 9, 15 | sstrd 3987 | . . 3 ⊢ (𝜑 → (𝐴 / ∼ ) ⊆ Fin) |
17 | qsdisj2 8788 | . . . 4 ⊢ ( ∼ Er 𝐴 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) | |
18 | 1, 17 | syl 17 | . . 3 ⊢ (𝜑 → Disj 𝑥 ∈ (𝐴 / ∼ )𝑥) |
19 | 10, 16, 18 | hashuni 15776 | . 2 ⊢ (𝜑 → (♯‘∪ (𝐴 / ∼ )) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
20 | 6, 19 | eqtr3d 2768 | 1 ⊢ (𝜑 → (♯‘𝐴) = Σ𝑥 ∈ (𝐴 / ∼ )(♯‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⊆ wss 3943 𝒫 cpw 4597 ∪ cuni 4902 Disj wdisj 5106 ‘cfv 6536 Er wer 8699 / cqs 8701 Fincfn 8938 ♯chash 14293 Σcsu 15636 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-ec 8704 df-qs 8708 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 |
This theorem is referenced by: lagsubg2 19118 sylow1lem3 19518 sylow2a 19537 hashclwwlkn0 29832 |
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