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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iblsplitf | Structured version Visualization version GIF version |
Description: A version of iblsplit 45623 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 2892 | . . 3 ⊢ Ⅎ𝑦𝐶 | |
2 | nfcsb1v 3916 | . . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 | |
3 | csbeq1a 3905 | . . 3 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) | |
4 | 1, 2, 3 | cbvmpt 5256 | . 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → 𝑦 ∈ 𝑈) | |
8 | iblsplitf.X | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
9 | nfv 1910 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑈 | |
10 | 8, 9 | nfan 1895 | . . . . 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 411 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → (𝑥 ∈ 𝑈 → 𝐶 ∈ ℂ)) |
14 | 10, 13 | ralrimi 3245 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ∀𝑥 ∈ 𝑈 𝐶 ∈ ℂ) |
15 | rspcsbela 4432 | . . . 4 ⊢ ((𝑦 ∈ 𝑈 ∧ ∀𝑥 ∈ 𝑈 𝐶 ∈ ℂ) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) | |
16 | 7, 14, 15 | syl2anc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑈) → ⦋𝑦 / 𝑥⦌𝐶 ∈ ℂ) |
17 | 3 | equcoms 2016 | . . . . . 6 ⊢ (𝑦 = 𝑥 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) |
18 | 17 | eqcomd 2732 | . . . . 5 ⊢ (𝑦 = 𝑥 → ⦋𝑦 / 𝑥⦌𝐶 = 𝐶) |
19 | 2, 1, 18 | cbvmpt 5256 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
20 | iblsplitf.a | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ 𝐿1) | |
21 | 19, 20 | eqeltrid 2830 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
22 | 2, 1, 18 | cbvmpt 5256 | . . . 4 ⊢ (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶) |
23 | iblsplitf.b | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ 𝐿1) | |
24 | 22, 23 | eqeltrid 2830 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
25 | 5, 6, 16, 21, 24 | iblsplit 45623 | . 2 ⊢ (𝜑 → (𝑦 ∈ 𝑈 ↦ ⦋𝑦 / 𝑥⦌𝐶) ∈ 𝐿1) |
26 | 4, 25 | eqeltrid 2830 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ 𝐶) ∈ 𝐿1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 ∀wral 3051 ⦋csb 3891 ∪ cun 3944 ∩ cin 3945 ↦ cmpt 5228 ‘cfv 6546 ℂcc 11147 0cc0 11149 vol*covol 25479 𝐿1cibl 25634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-disj 5111 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8726 df-map 8849 df-pm 8850 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-dju 9937 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13140 df-xadd 13141 df-xmul 13142 df-ioo 13376 df-ico 13378 df-icc 13379 df-fz 13533 df-fzo 13676 df-fl 13806 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 df-rest 17432 df-topgen 17453 df-psmet 21331 df-xmet 21332 df-met 21333 df-bl 21334 df-mopn 21335 df-top 22884 df-topon 22901 df-bases 22937 df-cmp 23379 df-ovol 25481 df-vol 25482 df-mbf 25636 df-itg1 25637 df-itg2 25638 df-ibl 25639 |
This theorem is referenced by: iblspltprt 45630 |
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