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Mirrors > Home > MPE Home > Th. List > ltexpri | Structured version Visualization version GIF version |
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltexpri | ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelpr 11023 | . . 3 ⊢ <P ⊆ (P × P) | |
2 | 1 | brel 5743 | . 2 ⊢ (𝐴<P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P)) |
3 | ltprord 11055 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
4 | oveq2 7427 | . . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑤 +Q 𝑦) = (𝑤 +Q 𝑧)) | |
5 | 4 | eleq1d 2810 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑤 +Q 𝑦) ∈ 𝐵 ↔ (𝑤 +Q 𝑧) ∈ 𝐵)) |
6 | 5 | anbi2d 628 | . . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ (¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵))) |
7 | 6 | exbidv 1916 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵))) |
8 | 7 | cbvabv 2798 | . . . . . . 7 ⊢ {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} = {𝑧 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)} |
9 | 8 | ltexprlem5 11065 | . . . . . 6 ⊢ ((𝐵 ∈ P ∧ 𝐴 ⊊ 𝐵) → {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P) |
10 | 9 | adantll 712 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P) |
11 | 8 | ltexprlem6 11066 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) ⊆ 𝐵) |
12 | 8 | ltexprlem7 11067 | . . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)})) |
13 | 11, 12 | eqssd 3994 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵) |
14 | oveq2 7427 | . . . . . . 7 ⊢ (𝑥 = {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → (𝐴 +P 𝑥) = (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)})) | |
15 | 14 | eqeq1d 2727 | . . . . . 6 ⊢ (𝑥 = {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵)) |
16 | 15 | rspcev 3606 | . . . . 5 ⊢ (({𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P ∧ (𝐴 +P {𝑦 ∣ ∃𝑤(¬ 𝑤 ∈ 𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵) → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
17 | 10, 13, 16 | syl2anc 582 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
18 | 17 | ex 411 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ⊊ 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)) |
19 | 3, 18 | sylbid 239 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵)) |
20 | 2, 19 | mpcom 38 | 1 ⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 ∃wrex 3059 ⊊ wpss 3945 class class class wbr 5149 (class class class)co 7419 +Q cplq 10880 Pcnp 10884 +P cpp 10886 <P cltp 10888 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-omul 8492 df-er 8725 df-ni 10897 df-pli 10898 df-mi 10899 df-lti 10900 df-plpq 10933 df-mpq 10934 df-ltpq 10935 df-enq 10936 df-nq 10937 df-erq 10938 df-plq 10939 df-mq 10940 df-1nq 10941 df-rq 10942 df-ltnq 10943 df-np 11006 df-plp 11008 df-ltp 11010 |
This theorem is referenced by: ltaprlem 11069 recexsrlem 11128 mulgt0sr 11130 map2psrpr 11135 |
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