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Mirrors > Home > ILE Home > Th. List > addextpr | GIF version |
Description: Strong extensionality of addition (ordering version). This is similar to addext 7777 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.) |
Ref | Expression |
---|---|
addextpr | ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶 ∨ 𝐵<P 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addclpr 6789 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P) | |
2 | 1 | adantr 270 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐴 +P 𝐵) ∈ P) |
3 | addclpr 6789 | . . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐷 ∈ P) → (𝐶 +P 𝐷) ∈ P) | |
4 | 3 | adantl 271 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐶 +P 𝐷) ∈ P) |
5 | simprl 498 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 𝐶 ∈ P) | |
6 | simplr 497 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 𝐵 ∈ P) | |
7 | addclpr 6789 | . . . 4 ⊢ ((𝐶 ∈ P ∧ 𝐵 ∈ P) → (𝐶 +P 𝐵) ∈ P) | |
8 | 5, 6, 7 | syl2anc 403 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐶 +P 𝐵) ∈ P) |
9 | ltsopr 6848 | . . . 4 ⊢ <P Or P | |
10 | sowlin 4083 | . . . 4 ⊢ ((<P Or P ∧ ((𝐴 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐷) ∈ P ∧ (𝐶 +P 𝐵) ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐵) ∨ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷)))) | |
11 | 9, 10 | mpan 415 | . . 3 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝐶 +P 𝐷) ∈ P ∧ (𝐶 +P 𝐵) ∈ P) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐵) ∨ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷)))) |
12 | 2, 4, 8, 11 | syl3anc 1170 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐵) ∨ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷)))) |
13 | simpll 496 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 𝐴 ∈ P) | |
14 | ltaprg 6871 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐶 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P 𝐶 ↔ (𝐵 +P 𝐴)<P (𝐵 +P 𝐶))) | |
15 | 13, 5, 6, 14 | syl3anc 1170 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐴<P 𝐶 ↔ (𝐵 +P 𝐴)<P (𝐵 +P 𝐶))) |
16 | addcomprg 6830 | . . . . . . 7 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓)) | |
17 | 16 | adantl 271 | . . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓)) |
18 | 17, 13, 6 | caovcomd 5688 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐴 +P 𝐵) = (𝐵 +P 𝐴)) |
19 | 17, 5, 6 | caovcomd 5688 | . . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐶 +P 𝐵) = (𝐵 +P 𝐶)) |
20 | 18, 19 | breq12d 3806 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐵) ↔ (𝐵 +P 𝐴)<P (𝐵 +P 𝐶))) |
21 | 15, 20 | bitr4d 189 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐴<P 𝐶 ↔ (𝐴 +P 𝐵)<P (𝐶 +P 𝐵))) |
22 | simprr 499 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → 𝐷 ∈ P) | |
23 | ltaprg 6871 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐷 ∈ P ∧ 𝐶 ∈ P) → (𝐵<P 𝐷 ↔ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷))) | |
24 | 6, 22, 5, 23 | syl3anc 1170 | . . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (𝐵<P 𝐷 ↔ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷))) |
25 | 21, 24 | orbi12d 740 | . 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴<P 𝐶 ∨ 𝐵<P 𝐷) ↔ ((𝐴 +P 𝐵)<P (𝐶 +P 𝐵) ∨ (𝐶 +P 𝐵)<P (𝐶 +P 𝐷)))) |
26 | 12, 25 | sylibrd 167 | 1 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ((𝐴 +P 𝐵)<P (𝐶 +P 𝐷) → (𝐴<P 𝐶 ∨ 𝐵<P 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3793 Or wor 4058 (class class class)co 5543 Pcnp 6543 +P cpp 6545 <P cltp 6547 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-coll 3901 ax-sep 3904 ax-nul 3912 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-iinf 4337 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-csb 2910 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-nul 3259 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-iun 3688 df-br 3794 df-opab 3848 df-mpt 3849 df-tr 3884 df-eprel 4052 df-id 4056 df-po 4059 df-iso 4060 df-iord 4129 df-on 4131 df-suc 4134 df-iom 4340 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-f1 4937 df-fo 4938 df-f1o 4939 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-1st 5798 df-2nd 5799 df-recs 5954 df-irdg 6019 df-1o 6065 df-2o 6066 df-oadd 6069 df-omul 6070 df-er 6172 df-ec 6174 df-qs 6178 df-ni 6556 df-pli 6557 df-mi 6558 df-lti 6559 df-plpq 6596 df-mpq 6597 df-enq 6599 df-nqqs 6600 df-plqqs 6601 df-mqqs 6602 df-1nqqs 6603 df-rq 6604 df-ltnqqs 6605 df-enq0 6676 df-nq0 6677 df-0nq0 6678 df-plq0 6679 df-mq0 6680 df-inp 6718 df-iplp 6720 df-iltp 6722 |
This theorem is referenced by: mulextsr1lem 7018 |
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