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Mirrors > Home > ILE Home > Th. List > caucvgprprlemk | GIF version |
Description: Lemma for caucvgprpr 7710. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 28-Nov-2020.) |
Ref | Expression |
---|---|
caucvgprprlemk.jk | ⊢ (𝜑 → 𝐽 <N 𝐾) |
caucvgprprlemk.jkq | ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) |
Ref | Expression |
---|---|
caucvgprprlemk | ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprprlemk.jk | . . . 4 ⊢ (𝜑 → 𝐽 <N 𝐾) | |
2 | ltrelpi 7322 | . . . . . 6 ⊢ <N ⊆ (N × N) | |
3 | 2 | brel 4678 | . . . . 5 ⊢ (𝐽 <N 𝐾 → (𝐽 ∈ N ∧ 𝐾 ∈ N)) |
4 | ltnnnq 7421 | . . . . 5 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) | |
5 | 1, 3, 4 | 3syl 17 | . . . 4 ⊢ (𝜑 → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) |
6 | 1, 5 | mpbid 147 | . . 3 ⊢ (𝜑 → [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q ) |
7 | ltrnqi 7419 | . . 3 ⊢ ([〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q → (*Q‘[〈𝐾, 1o〉] ~Q ) <Q (*Q‘[〈𝐽, 1o〉] ~Q )) | |
8 | ltnqpri 7592 | . . 3 ⊢ ((*Q‘[〈𝐾, 1o〉] ~Q ) <Q (*Q‘[〈𝐽, 1o〉] ~Q ) → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉) | |
9 | 6, 7, 8 | 3syl 17 | . 2 ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉) |
10 | caucvgprprlemk.jkq | . 2 ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) | |
11 | ltsopr 7594 | . . 3 ⊢ <P Or P | |
12 | ltrelpr 7503 | . . 3 ⊢ <P ⊆ (P × P) | |
13 | 11, 12 | sotri 5024 | . 2 ⊢ ((〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉 ∧ 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐽, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐽, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) |
14 | 9, 10, 13 | syl2anc 411 | 1 ⊢ (𝜑 → 〈{𝑙 ∣ 𝑙 <Q (*Q‘[〈𝐾, 1o〉] ~Q )}, {𝑢 ∣ (*Q‘[〈𝐾, 1o〉] ~Q ) <Q 𝑢}〉<P 𝑄) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 {cab 2163 〈cop 3595 class class class wbr 4003 ‘cfv 5216 1oc1o 6409 [cec 6532 Ncnpi 7270 <N clti 7273 ~Q ceq 7277 *Qcrq 7282 <Q cltq 7283 Pcnp 7289 <P cltp 7293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-mpq 7343 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-mqqs 7348 df-1nqqs 7349 df-rq 7350 df-ltnqqs 7351 df-inp 7464 df-iltp 7468 |
This theorem is referenced by: caucvgprprlem1 7707 caucvgprprlem2 7708 |
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