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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemk41 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.) |
Ref | Expression |
---|---|
cdlemk41.y | ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) |
Ref | Expression |
---|---|
cdlemk41 | ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvd 2898 | . 2 ⊢ (𝐺 ∈ 𝑇 → Ⅎ𝑔((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))) | |
2 | cdlemk41.y | . . 3 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) | |
3 | fveq2 6885 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘𝑔) = (𝑅‘𝐺)) | |
4 | 3 | oveq2d 7421 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑅‘𝐺))) |
5 | coeq1 5851 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 ∘ ◡𝑏) = (𝐺 ∘ ◡𝑏)) | |
6 | 5 | fveq2d 6889 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘(𝑔 ∘ ◡𝑏)) = (𝑅‘(𝐺 ∘ ◡𝑏))) |
7 | 6 | oveq2d 7421 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏))) = (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏)))) |
8 | 4, 7 | oveq12d 7423 | . . 3 ⊢ (𝑔 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ◡𝑏)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))) |
9 | 2, 8 | eqtrid 2778 | . 2 ⊢ (𝑔 = 𝐺 → 𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))) |
10 | 1, 9 | csbiegf 3922 | 1 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ◡𝑏))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⦋csb 3888 ◡ccnv 5668 ∘ ccom 5673 ‘cfv 6537 (class class class)co 7405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-co 5678 df-iota 6489 df-fv 6545 df-ov 7408 |
This theorem is referenced by: cdlemkid2 40308 cdlemkfid3N 40309 cdlemky 40310 cdlemk42yN 40328 |
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