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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 3510 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
3 | fveq2 6663 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾)) | |
4 | pltval.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | syl6eqr 2871 | . . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ ) |
6 | 5 | difeq1d 4095 | . . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I )) |
7 | df-plt 17556 | . . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) | |
8 | 4 | fvexi 6677 | . . . . 5 ⊢ ≤ ∈ V |
9 | 8 | difexi 5223 | . . . 4 ⊢ ( ≤ ∖ I ) ∈ V |
10 | 6, 7, 9 | fvmpt 6761 | . . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I )) |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I )) |
12 | 1, 11 | syl5eq 2865 | 1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 I cid 5452 ‘cfv 6348 lecple 16560 ltcplt 17539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-plt 17556 |
This theorem is referenced by: pltval 17558 opsrtoslem2 20193 relt 20687 oppglt 30568 xrslt 30590 submarchi 30742 |
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