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Mirrors > Home > MPE Home > Th. List > dvdsext | Structured version Visualization version 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 5106 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) | |
2 | 1 | ralrimivw 3145 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) |
3 | simpll 765 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∈ ℕ0) | |
4 | simplr 767 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∈ ℕ0) | |
5 | nn0z 12456 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
6 | iddvds 16086 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∥ 𝐵) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∥ 𝐵) |
8 | 7 | ad2antlr 725 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐵) |
9 | breq2 5107 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐵)) | |
10 | breq2 5107 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐵)) | |
11 | 9, 10 | bibi12d 345 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) ↔ (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵))) |
12 | 11 | rspcva 3577 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
13 | 12 | adantll 712 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
14 | 8, 13 | mpbird 256 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐵) |
15 | nn0z 12456 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
16 | iddvds 16086 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 𝐴) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∥ 𝐴) |
18 | 17 | ad2antrr 724 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐴) |
19 | breq2 5107 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐴)) | |
20 | breq2 5107 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐵 ∥ 𝑥 ↔ 𝐵 ∥ 𝐴)) | |
21 | 19, 20 | bibi12d 345 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) ↔ (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴))) |
22 | 21 | rspcva 3577 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
23 | 22 | adantlr 713 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
24 | 18, 23 | mpbid 231 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐴) |
25 | dvdseq 16130 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴)) → 𝐴 = 𝐵) | |
26 | 3, 4, 14, 24, 25 | syl22anc 837 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 = 𝐵) |
27 | 26 | ex 413 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) → 𝐴 = 𝐵)) |
28 | 2, 27 | impbid2 225 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 class class class wbr 5103 ℕ0cn0 12346 ℤcz 12432 ∥ cdvds 16070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 ax-pre-sup 11062 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-sup 9311 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-3 12150 df-n0 12347 df-z 12433 df-uz 12696 df-rp 12844 df-seq 13835 df-exp 13896 df-cj 14917 df-re 14918 df-im 14919 df-sqrt 15053 df-abs 15054 df-dvds 16071 |
This theorem is referenced by: odmulg 19269 znchr 20892 |
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