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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdleme31sn1c | Structured version Visualization version GIF version |
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 1-Mar-2013.) |
Ref | Expression |
---|---|
cdleme31sn1c.g | ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) |
cdleme31sn1c.i | ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) |
cdleme31sn1c.n | ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) |
cdleme31sn1c.y | ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) |
cdleme31sn1c.c | ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)) |
Ref | Expression |
---|---|
cdleme31sn1c | ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdleme31sn1c.i | . . 3 ⊢ 𝐼 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐺)) | |
2 | cdleme31sn1c.n | . . 3 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐷) | |
3 | eqid 2610 | . . 3 ⊢ (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) | |
4 | 1, 2, 3 | cdleme31sn1 34687 | . 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺))) |
5 | cdleme31sn1c.g | . . . . . . . . 9 ⊢ 𝐺 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊))) | |
6 | cdleme31sn1c.y | . . . . . . . . 9 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∧ (𝐸 ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) | |
7 | 5, 6 | cdleme31se 34688 | . . . . . . . 8 ⊢ (𝑅 ∈ 𝐴 → ⦋𝑅 / 𝑠⦌𝐺 = 𝑌) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝐺 = 𝑌) |
9 | 8 | eqeq2d 2620 | . . . . . 6 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (𝑦 = ⦋𝑅 / 𝑠⦌𝐺 ↔ 𝑦 = 𝑌)) |
10 | 9 | imbi2d 329 | . . . . 5 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) ↔ ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))) |
11 | 10 | ralbidv 2969 | . . . 4 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺) ↔ ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))) |
12 | 11 | riotabidv 6513 | . . 3 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌))) |
13 | cdleme31sn1c.c | . . 3 ⊢ 𝐶 = (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝑌)) | |
14 | 12, 13 | syl6eqr 2662 | . 2 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = ⦋𝑅 / 𝑠⦌𝐺)) = 𝐶) |
15 | 4, 14 | eqtrd 2644 | 1 ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ⦋𝑅 / 𝑠⦌𝑁 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⦋csb 3499 ifcif 4036 class class class wbr 4583 ℩crio 6510 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-riota 6511 df-ov 6552 |
This theorem is referenced by: cdlemefs32sn1aw 34720 cdleme43fsv1snlem 34726 cdleme41sn3a 34739 cdleme40m 34773 cdleme40n 34774 |
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