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Mirrors > Home > MPE Home > Th. List > hashssdif | Structured version Visualization version GIF version |
Description: The size of the difference of a finite set and a subset is the set's size minus the subset's. (Contributed by Steve Rodriguez, 24-Oct-2015.) |
Ref | Expression |
---|---|
hashssdif | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘(𝐴 ∖ 𝐵)) = ((#‘𝐴) − (#‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfi 8065 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
2 | diffi 8077 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin) | |
3 | disjdif 3992 | . . . . . . . 8 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅ | |
4 | hashun 13032 | . . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∖ 𝐵) ∈ Fin ∧ (𝐵 ∩ (𝐴 ∖ 𝐵)) = ∅) → (#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵)))) | |
5 | 3, 4 | mp3an3 1405 | . . . . . . 7 ⊢ ((𝐵 ∈ Fin ∧ (𝐴 ∖ 𝐵) ∈ Fin) → (#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵)))) |
6 | 1, 2, 5 | syl2an 493 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝐴 ∈ Fin) → (#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵)))) |
7 | 6 | anabss1 851 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵)))) |
8 | undif 4001 | . . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) | |
9 | 8 | biimpi 205 | . . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∪ (𝐴 ∖ 𝐵)) = 𝐴) |
10 | 9 | fveq2d 6107 | . . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = (#‘𝐴)) |
11 | 10 | eqeq1d 2612 | . . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ((#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) ↔ (#‘𝐴) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))))) |
12 | 11 | adantl 481 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ((#‘(𝐵 ∪ (𝐴 ∖ 𝐵))) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) ↔ (#‘𝐴) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))))) |
13 | 7, 12 | mpbid 221 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘𝐴) = ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵)))) |
14 | 13 | eqcomd 2616 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) = (#‘𝐴)) |
15 | hashcl 13009 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0) | |
16 | 15 | nn0cnd 11230 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℂ) |
17 | hashcl 13009 | . . . . . . . 8 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0) | |
18 | 1, 17 | syl 17 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘𝐵) ∈ ℕ0) |
19 | 18 | nn0cnd 11230 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘𝐵) ∈ ℂ) |
20 | hashcl 13009 | . . . . . . . 8 ⊢ ((𝐴 ∖ 𝐵) ∈ Fin → (#‘(𝐴 ∖ 𝐵)) ∈ ℕ0) | |
21 | 2, 20 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ Fin → (#‘(𝐴 ∖ 𝐵)) ∈ ℕ0) |
22 | 21 | nn0cnd 11230 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (#‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
23 | subadd 10163 | . . . . . 6 ⊢ (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ ∧ (#‘(𝐴 ∖ 𝐵)) ∈ ℂ) → (((#‘𝐴) − (#‘𝐵)) = (#‘(𝐴 ∖ 𝐵)) ↔ ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) = (#‘𝐴))) | |
24 | 16, 19, 22, 23 | syl3an 1360 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) ∧ 𝐴 ∈ Fin) → (((#‘𝐴) − (#‘𝐵)) = (#‘(𝐴 ∖ 𝐵)) ↔ ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) = (#‘𝐴))) |
25 | 24 | 3anidm13 1376 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴)) → (((#‘𝐴) − (#‘𝐵)) = (#‘(𝐴 ∖ 𝐵)) ↔ ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) = (#‘𝐴))) |
26 | 25 | anabss5 853 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (((#‘𝐴) − (#‘𝐵)) = (#‘(𝐴 ∖ 𝐵)) ↔ ((#‘𝐵) + (#‘(𝐴 ∖ 𝐵))) = (#‘𝐴))) |
27 | 14, 26 | mpbird 246 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → ((#‘𝐴) − (#‘𝐵)) = (#‘(𝐴 ∖ 𝐵))) |
28 | 27 | eqcomd 2616 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → (#‘(𝐴 ∖ 𝐵)) = ((#‘𝐴) − (#‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 + caddc 9818 − cmin 10145 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-hash 12980 |
This theorem is referenced by: hashdif 13062 hashdifsn 13063 brfi1indlem 13133 uvtxnm1nbgra 26022 nbhashuvtx1 26442 clwlknclwlkdifnum 26488 ballotlemfmpn 29883 ballotth 29926 poimirlem26 32605 poimirlem27 32606 uvtxanm1nbgr 40631 clwwlknclwwlkdifnum 41182 |
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