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| Mirrors > Home > MPE Home > Th. List > pltfval | Structured version Visualization version GIF version | ||
| Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.) |
| Ref | Expression |
|---|---|
| pltval.l | ⊢ ≤ = (le‘𝐾) |
| pltval.s | ⊢ < = (lt‘𝐾) |
| Ref | Expression |
|---|---|
| pltfval | ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pltval.s | . 2 ⊢ < = (lt‘𝐾) | |
| 2 | elex 3501 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
| 3 | fveq2 6906 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾)) | |
| 4 | pltval.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ ) |
| 6 | 5 | difeq1d 4125 | . . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I )) |
| 7 | df-plt 18375 | . . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) | |
| 8 | 4 | fvexi 6920 | . . . . 5 ⊢ ≤ ∈ V |
| 9 | 8 | difexi 5330 | . . . 4 ⊢ ( ≤ ∖ I ) ∈ V |
| 10 | 6, 7, 9 | fvmpt 7016 | . . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I )) |
| 11 | 2, 10 | syl 17 | . 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I )) |
| 12 | 1, 11 | eqtrid 2789 | 1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 I cid 5577 ‘cfv 6561 lecple 17304 ltcplt 18354 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 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-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-plt 18375 |
| This theorem is referenced by: pltval 18377 relt 21633 opsrtoslem2 22080 oppglt 32953 xrslt 33009 submarchi 33193 |
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