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Mirrors > Home > ILE Home > Th. List > caovdilemd | GIF version |
Description: Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.) |
Ref | Expression |
---|---|
caovdilemd.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
caovdilemd.distr | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) |
caovdilemd.ass | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) |
caovdilemd.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆) |
caovdilemd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovdilemd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovdilemd.c | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovdilemd.d | ⊢ (𝜑 → 𝐷 ∈ 𝑆) |
caovdilemd.h | ⊢ (𝜑 → 𝐻 ∈ 𝑆) |
Ref | Expression |
---|---|
caovdilemd | ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdilemd.distr | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐹(𝑦𝐺𝑧))) | |
2 | caovdilemd.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) ∈ 𝑆) | |
3 | caovdilemd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | caovdilemd.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | 2, 3, 4 | caovcld 6049 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) ∈ 𝑆) |
6 | caovdilemd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
7 | caovdilemd.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
8 | 2, 6, 7 | caovcld 6049 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐷) ∈ 𝑆) |
9 | caovdilemd.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑆) | |
10 | 1, 5, 8, 9 | caovdird 6074 | . 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻))) |
11 | caovdilemd.ass | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))) | |
12 | 11, 3, 4, 9 | caovassd 6055 | . . 3 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻))) |
13 | 11, 6, 7, 9 | caovassd 6055 | . . 3 ⊢ (𝜑 → ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻))) |
14 | 12, 13 | oveq12d 5913 | . 2 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
15 | 10, 14 | eqtrd 2222 | 1 ⊢ (𝜑 → (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 (class class class)co 5895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 df-ov 5898 |
This theorem is referenced by: caovlem2d 6088 addassnqg 7410 addassnq0 7490 axmulass 7901 |
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