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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 3512 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
3 | fveq2 6664 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾)) | |
4 | pltval.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | syl6eqr 2874 | . . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ ) |
6 | 5 | difeq1d 4097 | . . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I )) |
7 | df-plt 17562 | . . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) | |
8 | 4 | fvexi 6678 | . . . . 5 ⊢ ≤ ∈ V |
9 | 8 | difexi 5224 | . . . 4 ⊢ ( ≤ ∖ I ) ∈ V |
10 | 6, 7, 9 | fvmpt 6762 | . . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I )) |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I )) |
12 | 1, 11 | syl5eq 2868 | 1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 I cid 5453 ‘cfv 6349 lecple 16566 ltcplt 17545 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-plt 17562 |
This theorem is referenced by: pltval 17564 opsrtoslem2 20259 relt 20753 oppglt 30636 xrslt 30658 submarchi 30810 |
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