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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31sn | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 26-Feb-2013.) |
Ref | Expression |
---|---|
cdleme31sn.n | ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) |
cdleme31sn.c | ⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) |
Ref | Expression |
---|---|
cdleme31sn | ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2010 | . . . . 5 ⊢ Ⅎ𝑠 𝑅 ≤ (𝑃 ∨ 𝑄) | |
2 | nfcsb1v 3742 | . . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐼 | |
3 | nfcsb1v 3742 | . . . . 5 ⊢ Ⅎ𝑠⦋𝑅 / 𝑠⦌𝐷 | |
4 | 1, 2, 3 | nfif 4304 | . . . 4 ⊢ Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑅 ∈ 𝐴 → Ⅎ𝑠if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)) |
6 | breq1 4844 | . . . 4 ⊢ (𝑠 = 𝑅 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑃 ∨ 𝑄))) | |
7 | csbeq1a 3735 | . . . 4 ⊢ (𝑠 = 𝑅 → 𝐼 = ⦋𝑅 / 𝑠⦌𝐼) | |
8 | csbeq1a 3735 | . . . 4 ⊢ (𝑠 = 𝑅 → 𝐷 = ⦋𝑅 / 𝑠⦌𝐷) | |
9 | 6, 7, 8 | ifbieq12d 4302 | . . 3 ⊢ (𝑠 = 𝑅 → if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)) |
10 | 5, 9 | csbiegf 3750 | . 2 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷)) |
11 | cdleme31sn.n | . . 3 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) | |
12 | 11 | csbeq2i 4186 | . 2 ⊢ ⦋𝑅 / 𝑠⦌𝑁 = ⦋𝑅 / 𝑠⦌if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) |
13 | cdleme31sn.c | . 2 ⊢ 𝐶 = if(𝑅 ≤ (𝑃 ∨ 𝑄), ⦋𝑅 / 𝑠⦌𝐼, ⦋𝑅 / 𝑠⦌𝐷) | |
14 | 10, 12, 13 | 3eqtr4g 2856 | 1 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Ⅎwnfc 2926 ⦋csb 3726 ifcif 4275 class class class wbr 4841 (class class class)co 6876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 |
This theorem is referenced by: cdleme31sn1 36394 cdleme31sn2 36402 |
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