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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 5151 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) | |
2 | 1 | ralrimivw 3148 | . 2 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) |
3 | simpll 767 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∈ ℕ0) | |
4 | simplr 769 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∈ ℕ0) | |
5 | nn0z 12636 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ) | |
6 | iddvds 16304 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∥ 𝐵) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∥ 𝐵) |
8 | 7 | ad2antlr 727 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐵) |
9 | breq2 5152 | . . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐵)) | |
10 | breq2 5152 | . . . . . . . 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 3620 | . . . . . 6 ⊢ ((𝐵 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
13 | 12 | adantll 714 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐵 ↔ 𝐵 ∥ 𝐵)) |
14 | 8, 13 | mpbird 257 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐵) |
15 | nn0z 12636 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
16 | iddvds 16304 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 𝐴 ∥ 𝐴) | |
17 | 15, 16 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∥ 𝐴) |
18 | 17 | ad2antrr 726 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 ∥ 𝐴) |
19 | breq2 5152 | . . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∥ 𝑥 ↔ 𝐴 ∥ 𝐴)) | |
20 | breq2 5152 | . . . . . . . 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 3620 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ0 ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
23 | 22 | adantlr 715 | . . . . 5 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → (𝐴 ∥ 𝐴 ↔ 𝐵 ∥ 𝐴)) |
24 | 18, 23 | mpbid 232 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐵 ∥ 𝐴) |
25 | dvdseq 16348 | . . . 4 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐴 ∥ 𝐵 ∧ 𝐵 ∥ 𝐴)) → 𝐴 = 𝐵) | |
26 | 3, 4, 14, 24, 25 | syl22anc 839 | . . 3 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥)) → 𝐴 = 𝐵) |
27 | 26 | ex 412 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥) → 𝐴 = 𝐵)) |
28 | 2, 27 | impbid2 226 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ ℕ0 (𝐴 ∥ 𝑥 ↔ 𝐵 ∥ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 class class class wbr 5148 ℕ0cn0 12524 ℤcz 12611 ∥ cdvds 16287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 |
This theorem is referenced by: odmulg 19589 znchr 21599 |
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