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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 3493 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V) | |
3 | fveq2 6881 | . . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾)) | |
4 | pltval.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
5 | 3, 4 | eqtr4di 2791 | . . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ ) |
6 | 5 | difeq1d 4119 | . . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I )) |
7 | df-plt 18270 | . . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I )) | |
8 | 4 | fvexi 6895 | . . . . 5 ⊢ ≤ ∈ V |
9 | 8 | difexi 5324 | . . . 4 ⊢ ( ≤ ∖ I ) ∈ V |
10 | 6, 7, 9 | fvmpt 6987 | . . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I )) |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I )) |
12 | 1, 11 | eqtrid 2785 | 1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3943 I cid 5569 ‘cfv 6535 lecple 17191 ltcplt 18248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 df-plt 18270 |
This theorem is referenced by: pltval 18272 relt 21141 opsrtoslem2 21585 oppglt 32103 xrslt 32148 submarchi 32303 |
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