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Mirrors > Home > ILE Home > Th. List > frec2uzlt2d | GIF version |
Description: The mapping 𝐺 (see frec2uz0d 9709) preserves order. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frec2uzzd.a | ⊢ (𝜑 → 𝐴 ∈ ω) |
frec2uzltd.b | ⊢ (𝜑 → 𝐵 ∈ ω) |
Ref | Expression |
---|---|
frec2uzlt2d | ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frec2uz.2 | . . 3 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
3 | frec2uzzd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ω) | |
4 | frec2uzltd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ω) | |
5 | 1, 2, 3, 4 | frec2uzltd 9713 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → (𝐺‘𝐴) < (𝐺‘𝐵))) |
6 | nntri3or 6189 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) | |
7 | 3, 4, 6 | syl2anc 403 | . . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)) |
8 | ax-1 5 | . . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵)) | |
9 | 8 | a1i 9 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵))) |
10 | fveq2 5256 | . . . . . . . . . 10 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
11 | 10 | adantl 271 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐺‘𝐴) = (𝐺‘𝐵)) |
12 | 11 | breq2d 3826 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐺‘𝐴) < (𝐺‘𝐴) ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
13 | 12 | biimpar 291 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ (𝐺‘𝐴) < (𝐺‘𝐵)) → (𝐺‘𝐴) < (𝐺‘𝐴)) |
14 | 1, 2, 3 | frec2uzzd 9710 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℤ) |
15 | 14 | adantr 270 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐺‘𝐴) ∈ ℤ) |
16 | 15 | adantr 270 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ (𝐺‘𝐴) < (𝐺‘𝐵)) → (𝐺‘𝐴) ∈ ℤ) |
17 | 16 | zred 8778 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ (𝐺‘𝐴) < (𝐺‘𝐵)) → (𝐺‘𝐴) ∈ ℝ) |
18 | 17 | ltnrd 7517 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ (𝐺‘𝐴) < (𝐺‘𝐵)) → ¬ (𝐺‘𝐴) < (𝐺‘𝐴)) |
19 | 13, 18 | pm2.21dd 583 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐴 = 𝐵) ∧ (𝐺‘𝐴) < (𝐺‘𝐵)) → 𝐴 ∈ 𝐵) |
20 | 19 | ex 113 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵)) |
21 | 20 | ex 113 | . . . 4 ⊢ (𝜑 → (𝐴 = 𝐵 → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵))) |
22 | 1, 2, 4 | frec2uzzd 9710 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℤ) |
23 | 22 | adantr 270 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → (𝐺‘𝐵) ∈ ℤ) |
24 | 23 | zred 8778 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → (𝐺‘𝐵) ∈ ℝ) |
25 | 14 | adantr 270 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → (𝐺‘𝐴) ∈ ℤ) |
26 | 25 | zred 8778 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → (𝐺‘𝐴) ∈ ℝ) |
27 | 1, 2, 4, 3 | frec2uzltd 9713 | . . . . . . . 8 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → (𝐺‘𝐵) < (𝐺‘𝐴))) |
28 | 27 | imp 122 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → (𝐺‘𝐵) < (𝐺‘𝐴)) |
29 | 24, 26, 28 | ltnsymd 7524 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → ¬ (𝐺‘𝐴) < (𝐺‘𝐵)) |
30 | 29 | pm2.21d 582 | . . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝐴) → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵)) |
31 | 30 | ex 113 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ 𝐴 → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵))) |
32 | 9, 21, 31 | 3jaod 1238 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵))) |
33 | 7, 32 | mpd 13 | . 2 ⊢ (𝜑 → ((𝐺‘𝐴) < (𝐺‘𝐵) → 𝐴 ∈ 𝐵)) |
34 | 5, 33 | impbid 127 | 1 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ↔ (𝐺‘𝐴) < (𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∨ w3o 921 = wceq 1287 ∈ wcel 1436 class class class wbr 3814 ↦ cmpt 3868 ωcom 4371 ‘cfv 4972 (class class class)co 5594 freccfrec 6090 1c1 7272 + caddc 7274 < clt 7443 ℤcz 8660 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-recs 6005 df-frec 6091 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 |
This theorem is referenced by: frec2uzisod 9717 frec2uzled 9739 |
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