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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblsplitf | Structured version Visualization version GIF version |
Description: A version of iblsplit 42258 using bound-variable hypotheses instead of distinct variable conditions" (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
iblsplitf.X | ⊢ Ⅎ𝑥𝜑 |
iblsplitf.vol | ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) |
iblsplitf.u | ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) |
iblsplitf.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ) |
iblsplitf.a | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) |
iblsplitf.b | ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1) |
Ref | Expression |
---|---|
iblsplitf | ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2979 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
2 | nfcsb1v 3909 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
3 | csbeq1a 3899 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
4 | 1, 2, 3 | cbvmpt 5169 | . 2 ⊢ (𝑥 ∈ 𝑈 ↦ 𝐶) = (𝑦 ∈ 𝑈 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
5 | iblsplitf.vol | . . 3 ⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) = 0) | |
6 | iblsplitf.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐴 ∪ 𝐵)) | |
7 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
8 | iblsplitf.X | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
9 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑈 | |
10 | 8, 9 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝑈) |
11 | iblsplitf.c | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ) | |
12 | 11 | adantlr 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑈) ∧ 𝑥 ∈ 𝑈) → 𝐶 ∈ ℂ) |
13 | 12 | ex 415 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → (𝑥 ∈ 𝑈 → 𝐶 ∈ ℂ)) |
14 | 10, 13 | ralrimi 3218 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 𝐶 ∈ ℂ) |
15 | rspcsbela 4389 | . . . 4 ⊢ ((𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 𝐶 ∈ ℂ) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) | |
16 | 7, 14, 15 | syl2anc 586 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) |
17 | 3 | equcoms 2027 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) |
18 | 17 | eqcomd 2829 | . . . . 5 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) |
19 | 2, 1, 18 | cbvmpt 5169 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
20 | iblsplitf.a | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) | |
21 | 19, 20 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
22 | 2, 1, 18 | cbvmpt 5169 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) |
23 | iblsplitf.b | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1) | |
24 | 22, 23 | eqeltrid 2919 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
25 | 5, 6, 16, 21, 24 | iblsplit 42258 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
26 | 4, 25 | eqeltrid 2919 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3140 ⦋csb 3885 ∪ cun 3936 ∩ cin 3937 ↦ cmpt 5148 ‘cfv 6357 ℂcc 10537 0cc0 10539 vol*covol 24065 𝐿1cibl 24220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-disj 5034 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-ofr 7412 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-rest 16698 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cmp 21997 df-ovol 24067 df-vol 24068 df-mbf 24222 df-itg1 24223 df-itg2 24224 df-ibl 24225 |
This theorem is referenced by: iblspltprt 42265 |
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