![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ismntd | Structured version Visualization version GIF version |
Description: Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.) |
Ref | Expression |
---|---|
ismntd.1 | ⊢ 𝐴 = (Base‘𝑉) |
ismntd.2 | ⊢ 𝐵 = (Base‘𝑊) |
ismntd.3 | ⊢ ≤ = (le‘𝑉) |
ismntd.4 | ⊢ ≲ = (le‘𝑊) |
ismntd.5 | ⊢ (𝜑 → 𝑉 ∈ 𝐶) |
ismntd.6 | ⊢ (𝜑 → 𝑊 ∈ 𝐷) |
ismntd.7 | ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) |
ismntd.8 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
ismntd.9 | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
ismntd.10 | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
Ref | Expression |
---|---|
ismntd | ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismntd.5 | . . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐶) | |
2 | ismntd.6 | . . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐷) | |
3 | ismntd.7 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊)) | |
4 | ismntd.1 | . . . . . 6 ⊢ 𝐴 = (Base‘𝑉) | |
5 | ismntd.2 | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
6 | ismntd.3 | . . . . . 6 ⊢ ≤ = (le‘𝑉) | |
7 | ismntd.4 | . . . . . 6 ⊢ ≲ = (le‘𝑊) | |
8 | 4, 5, 6, 7 | ismnt 30772 | . . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))) |
9 | 8 | biimp3a 1467 | . . . 4 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))) |
10 | 9 | simprd 500 | . . 3 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) |
11 | 1, 2, 3, 10 | syl3anc 1369 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) |
12 | ismntd.10 | . 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
13 | breq1 5028 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦)) | |
14 | fveq2 6651 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
15 | 14 | breq1d 5035 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑦))) |
16 | 13, 15 | imbi12d 349 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)))) |
17 | breq2 5029 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌)) | |
18 | fveq2 6651 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
19 | 18 | breq2d 5037 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌))) |
20 | 17, 19 | imbi12d 349 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)))) |
21 | ismntd.8 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
22 | eqidd 2760 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴) | |
23 | ismntd.9 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
24 | 16, 20, 21, 22, 23 | rspc2vd 3850 | . 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)))) |
25 | 11, 12, 24 | mp2d 49 | 1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 ∀wral 3068 class class class wbr 5025 ⟶wf 6324 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 lecple 16615 Monotcmnt 30767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ral 3073 df-rex 3074 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-opab 5088 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-map 8411 df-mnt 30769 |
This theorem is referenced by: mgcmntco 30783 mgcf1o 30792 |
Copyright terms: Public domain | W3C validator |