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Mirrors > Home > MPE Home > Th. List > itg1addlem1 | Structured version Visualization version GIF version |
Description: Decompose a preimage, which is always a disjoint union. (Contributed by Mario Carneiro, 25-Jun-2014.) (Proof shortened by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
itg1addlem.1 | ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
itg1addlem.2 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
itg1addlem.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ (◡𝐹 “ {𝑘})) |
itg1addlem.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ dom vol) |
itg1addlem.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ) |
Ref | Expression |
---|---|
itg1addlem1 | ⊢ (𝜑 → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg1addlem.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | itg1addlem.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ dom vol) | |
3 | itg1addlem.5 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (vol‘𝐵) ∈ ℝ) | |
4 | 2, 3 | jca 553 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)) |
5 | 4 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)) |
6 | itg1addlem.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ (◡𝐹 “ {𝑘})) | |
7 | 6 | adantrr 749 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → 𝐵 ⊆ (◡𝐹 “ {𝑘})) |
8 | simprr 792 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ 𝐵) | |
9 | 7, 8 | sseldd 3569 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → 𝑥 ∈ (◡𝐹 “ {𝑘})) |
10 | itg1addlem.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | |
11 | ffn 5958 | . . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → 𝐹 Fn 𝑋) | |
12 | 10, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋) |
13 | 12 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → 𝐹 Fn 𝑋) |
14 | fniniseg 6246 | . . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) = 𝑘))) | |
15 | 13, 14 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ (◡𝐹 “ {𝑘}) ↔ (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) = 𝑘))) |
16 | 9, 15 | mpbid 221 | . . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝑋 ∧ (𝐹‘𝑥) = 𝑘)) |
17 | 16 | simprd 478 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = 𝑘) |
18 | 17 | ralrimivva 2954 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑘) |
19 | invdisj 4571 | . . 3 ⊢ (∀𝑘 ∈ 𝐴 ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑘 → Disj 𝑘 ∈ 𝐴 𝐵) | |
20 | 18, 19 | syl 17 | . 2 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) |
21 | volfiniun 23122 | . 2 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)) | |
22 | 1, 5, 20, 21 | syl3anc 1318 | 1 ⊢ (𝜑 → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 {csn 4125 ∪ ciun 4455 Disj wdisj 4553 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 Fincfn 7841 ℝcr 9814 Σcsu 14264 volcvol 23039 |
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-inf2 8421 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 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 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-disj 4554 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-se 4998 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-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 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-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 |
This theorem is referenced by: itg1addlem4 23272 itg1addlem5 23273 |
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