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Mirrors > Home > MPE Home > Th. List > leisorel | Structured version Visualization version GIF version |
Description: Version of isorel 7058 for strictly increasing functions on the reals. (Contributed by Mario Carneiro, 6-Apr-2015.) (Revised by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
leisorel | ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leiso 13813 | . . . 4 ⊢ ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))) | |
2 | 1 | biimpcd 252 | . . 3 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))) |
3 | isorel 7058 | . . . 4 ⊢ ((𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))) | |
4 | 3 | ex 416 | . . 3 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷)))) |
5 | 2, 4 | syl6 35 | . 2 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) → ((𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))))) |
6 | 5 | 3imp 1108 | 1 ⊢ ((𝐹 Isom < , < (𝐴, 𝐵) ∧ (𝐴 ⊆ ℝ* ∧ 𝐵 ⊆ ℝ*) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶 ≤ 𝐷 ↔ (𝐹‘𝐶) ≤ (𝐹‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 ‘cfv 6324 Isom wiso 6325 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-le 10670 |
This theorem is referenced by: seqcoll 13818 isercolllem2 15014 isercoll 15016 summolem2a 15064 prodmolem2a 15280 xrhmeo 23551 fourierdlem52 42800 |
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