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Mirrors > Home > ILE Home > Th. List > dvdsext | GIF version |
Description: Poset extensionality for division. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
Ref | Expression |
---|---|
dvdsext | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3968 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) | |
2 | 1 | ralrimivw 2531 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) |
3 | simpll 519 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∈ ℕ0) | |
4 | simplr 520 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∈ ℕ0) | |
5 | nn0z 9187 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
6 | iddvds 11700 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∥ 𝐵) | |
7 | 5, 6 | syl 14 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∥ 𝐵) |
8 | 7 | ad2antlr 481 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐵) |
9 | breq2 3969 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐵)) | |
10 | breq2 3969 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐵)) | |
11 | 9, 10 | bibi12d 234 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) ↔ (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵))) |
12 | 11 | rspcva 2814 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
13 | 12 | adantll 468 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
14 | 8, 13 | mpbird 166 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐵) |
15 | nn0z 9187 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
16 | iddvds 11700 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 𝐴) | |
17 | 15, 16 | syl 14 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∥ 𝐴) |
18 | 17 | ad2antrr 480 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐴) |
19 | breq2 3969 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐴)) | |
20 | breq2 3969 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐴)) | |
21 | 19, 20 | bibi12d 234 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) ↔ (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴))) |
22 | 21 | rspcva 2814 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
23 | 22 | adantlr 469 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
24 | 18, 23 | mpbid 146 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐴) |
25 | dvdseq 11740 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴)) → 𝐴 = 𝐵) | |
26 | 3, 4, 14, 24, 25 | syl22anc 1221 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 = 𝐵) |
27 | 26 | ex 114 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) → 𝐴 = 𝐵)) |
28 | 2, 27 | impbid2 142 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 ∀wral 2435 class class class wbr 3965 ℕ0cn0 9090 ℤcz 9167 ∥ cdvds 11683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-q 9529 df-rp 9561 df-seqfrec 10345 df-exp 10419 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899 df-dvds 11684 |
This theorem is referenced by: (None) |
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