Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > monoord2xr | Structured version Visualization version GIF version |
Description: Ordering relation for a monotonic sequence, decreasing case. (Contributed by Glauco Siliprandi, 13-Feb-2022.) |
Ref | Expression |
---|---|
monoord2xr.p | ⊢ Ⅎ𝑘𝜑 |
monoord2xr.k | ⊢ Ⅎ𝑘𝐹 |
monoord2xr.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
monoord2xr.x | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) |
monoord2xr.l | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
monoord2xr | ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | monoord2xr.n | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | monoord2xr.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
3 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...𝑁) | |
4 | 2, 3 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) |
5 | monoord2xr.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
6 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
7 | 5, 6 | nffv 6680 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
8 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘ℝ* | |
9 | 7, 8 | nfel 2992 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℝ* |
10 | 4, 9 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
11 | eleq1w 2895 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁))) | |
12 | 11 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)))) |
13 | fveq2 6670 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
14 | 13 | eleq1d 2897 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ* ↔ (𝐹‘𝑗) ∈ ℝ*)) |
15 | 12, 14 | imbi12d 347 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*))) |
16 | monoord2xr.x | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ*) | |
17 | 10, 15, 16 | chvarfv 2242 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
18 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...(𝑁 − 1)) | |
19 | 2, 18 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) |
20 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
21 | 5, 20 | nffv 6680 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
22 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
23 | 21, 22, 7 | nfbr 5113 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗) |
24 | 19, 23 | nfim 1897 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
25 | eleq1w 2895 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...(𝑁 − 1)) ↔ 𝑗 ∈ (𝑀...(𝑁 − 1)))) | |
26 | 25 | anbi2d 630 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))))) |
27 | fvoveq1 7179 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) | |
28 | 27, 13 | breq12d 5079 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))) |
29 | 26, 28 | imbi12d 347 | . . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)))) |
30 | monoord2xr.l | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) | |
31 | 24, 29, 30 | chvarfv 2242 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
32 | 1, 17, 31 | monoord2xrv 41780 | 1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 ℝ*cxr 10674 ≤ cle 10676 − cmin 10870 ℤ≥cuz 12244 ...cfz 12893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-xneg 12508 df-fz 12894 |
This theorem is referenced by: (None) |
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