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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 1909 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...𝑁) | |
4 | 2, 3 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) |
5 | monoord2xr.k | . . . . . 6 ⊢ Ⅎ𝑘𝐹 | |
6 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑘𝑗 | |
7 | 5, 6 | nffv 6895 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗) |
8 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑘ℝ* | |
9 | 7, 8 | nfel 2911 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℝ* |
10 | 4, 9 | nfim 1891 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
11 | eleq1w 2810 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁))) | |
12 | 11 | anbi2d 628 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)))) |
13 | fveq2 6885 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗)) | |
14 | 13 | eleq1d 2812 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ* ↔ (𝐹‘𝑗) ∈ ℝ*)) |
15 | 12, 14 | imbi12d 344 | . . 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 2225 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ*) |
18 | nfv 1909 | . . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ (𝑀...(𝑁 − 1)) | |
19 | 2, 18 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) |
20 | nfcv 2897 | . . . . . 6 ⊢ Ⅎ𝑘(𝑗 + 1) | |
21 | 5, 20 | nffv 6895 | . . . . 5 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) |
22 | nfcv 2897 | . . . . 5 ⊢ Ⅎ𝑘 ≤ | |
23 | 21, 22, 7 | nfbr 5188 | . . . 4 ⊢ Ⅎ𝑘(𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗) |
24 | 19, 23 | nfim 1891 | . . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
25 | eleq1w 2810 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...(𝑁 − 1)) ↔ 𝑗 ∈ (𝑀...(𝑁 − 1)))) | |
26 | 25 | anbi2d 628 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))))) |
27 | fvoveq1 7428 | . . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝑗 + 1))) | |
28 | 27, 13 | breq12d 5154 | . . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘) ↔ (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗))) |
29 | 26, 28 | imbi12d 344 | . . 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 2225 | . 2 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑗 + 1)) ≤ (𝐹‘𝑗)) |
32 | 1, 17, 31 | monoord2xrv 44766 | 1 ⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝐹‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2877 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 1c1 11113 + caddc 11115 ℝ*cxr 11251 ≤ cle 11253 − cmin 11448 ℤ≥cuz 12826 ...cfz 13490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-xneg 13098 df-fz 13491 |
This theorem is referenced by: (None) |
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