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| Mirrors > Home > MPE Home > Th. List > Mathboxes > carsgmon | Structured version Visualization version GIF version | ||
| Description: Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.) |
| Ref | Expression |
|---|---|
| carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
| carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
| carsgmon.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| carsgmon.2 | ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂) |
| carsgmon.3 | ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) |
| Ref | Expression |
|---|---|
| carsgmon | ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | carsgmon.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂) | |
| 2 | carsgmon.1 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 3 | 1, 2 | ssexd 5324 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
| 4 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
| 5 | sseq1 4009 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦)) | |
| 6 | 5 | 3anbi2d 1443 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂))) |
| 7 | fveq2 6906 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑀‘𝑥) = (𝑀‘𝐴)) | |
| 8 | 7 | breq1d 5153 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑀‘𝑥) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝑦))) |
| 9 | 6, 8 | imbi12d 344 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝑦)))) |
| 10 | sseq2 4010 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵)) | |
| 11 | eleq1 2829 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂 ↔ 𝐵 ∈ 𝒫 𝑂)) | |
| 12 | 10, 11 | 3anbi23d 1441 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂))) |
| 13 | fveq2 6906 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑀‘𝑦) = (𝑀‘𝐵)) | |
| 14 | 13 | breq2d 5155 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑀‘𝐴) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝐵))) |
| 15 | 12, 14 | imbi12d 344 | . . . 4 ⊢ (𝑦 = 𝐵 → (((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝑦)) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)))) |
| 16 | carsgmon.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) | |
| 17 | 9, 15, 16 | vtocl2g 3574 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))) |
| 18 | 17 | imp 406 | . 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
| 19 | 3, 1, 4, 2, 1, 18 | syl23anc 1379 | 1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: carsggect 34320 carsgclctunlem2 34321 |
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