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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 3499 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
3 | fveq2 6907 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾)) | |
4 | pltval.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | eqtr4di 2793 | . . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ ) |
6 | 5 | difeq1d 4135 | . . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I )) |
7 | df-plt 18388 | . . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) | |
8 | 4 | fvexi 6921 | . . . . 5 ⊢ ≤ ∈ V |
9 | 8 | difexi 5336 | . . . 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 2787 | 1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 I cid 5582 ‘cfv 6563 lecple 17305 ltcplt 18366 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-plt 18388 |
This theorem is referenced by: pltval 18390 relt 21651 opsrtoslem2 22098 oppglt 32938 xrslt 32992 submarchi 33176 |
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