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Mirrors > Home > HSE Home > Th. List > 5oalem1 | Structured version Visualization version GIF version |
Description: Lemma for orthoarguesian law 5OA. (Contributed by NM, 1-Apr-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
5oalem1.1 | ⊢ 𝐴 ∈ Sℋ |
5oalem1.2 | ⊢ 𝐵 ∈ Sℋ |
5oalem1.3 | ⊢ 𝐶 ∈ Sℋ |
5oalem1.4 | ⊢ 𝑅 ∈ Sℋ |
Ref | Expression |
---|---|
5oalem1 | ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 813 | . . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ 𝐴) | |
2 | 5oalem1.1 | . . . . . . . 8 ⊢ 𝐴 ∈ Sℋ | |
3 | 2 | sheli 28199 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℋ) |
4 | 3 | ad2antrr 762 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → 𝑥 ∈ ℋ) |
5 | 5oalem1.3 | . . . . . . . 8 ⊢ 𝐶 ∈ Sℋ | |
6 | 5 | sheli 28199 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ ℋ) |
7 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅) → 𝑧 ∈ ℋ) |
8 | hvaddsub12 28023 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +ℎ (𝑧 −ℎ 𝑧)) = (𝑧 +ℎ (𝑥 −ℎ 𝑧))) | |
9 | 8 | 3anidm23 1425 | . . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +ℎ (𝑧 −ℎ 𝑧)) = (𝑧 +ℎ (𝑥 −ℎ 𝑧))) |
10 | hvsubid 28011 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℋ → (𝑧 −ℎ 𝑧) = 0ℎ) | |
11 | 10 | oveq2d 6706 | . . . . . . . 8 ⊢ (𝑧 ∈ ℋ → (𝑥 +ℎ (𝑧 −ℎ 𝑧)) = (𝑥 +ℎ 0ℎ)) |
12 | ax-hvaddid 27989 | . . . . . . . 8 ⊢ (𝑥 ∈ ℋ → (𝑥 +ℎ 0ℎ) = 𝑥) | |
13 | 11, 12 | sylan9eqr 2707 | . . . . . . 7 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑥 +ℎ (𝑧 −ℎ 𝑧)) = 𝑥) |
14 | 9, 13 | eqtr3d 2687 | . . . . . 6 ⊢ ((𝑥 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (𝑧 +ℎ (𝑥 −ℎ 𝑧)) = 𝑥) |
15 | 4, 7, 14 | syl2an 493 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → (𝑧 +ℎ (𝑥 −ℎ 𝑧)) = 𝑥) |
16 | 5oalem1.4 | . . . . . . 7 ⊢ 𝑅 ∈ Sℋ | |
17 | 5, 16 | shsvai 28351 | . . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅) → (𝑧 +ℎ (𝑥 −ℎ 𝑧)) ∈ (𝐶 +ℋ 𝑅)) |
18 | 17 | adantl 481 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → (𝑧 +ℎ (𝑥 −ℎ 𝑧)) ∈ (𝐶 +ℋ 𝑅)) |
19 | 15, 18 | eqeltrrd 2731 | . . . 4 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ (𝐶 +ℋ 𝑅)) |
20 | 1, 19 | elind 3831 | . . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑥 ∈ (𝐴 ∩ (𝐶 +ℋ 𝑅))) |
21 | simpllr 815 | . . 3 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑦 ∈ 𝐵) | |
22 | 5, 16 | shscli 28304 | . . . . . 6 ⊢ (𝐶 +ℋ 𝑅) ∈ Sℋ |
23 | 2, 22 | shincli 28349 | . . . . 5 ⊢ (𝐴 ∩ (𝐶 +ℋ 𝑅)) ∈ Sℋ |
24 | 5oalem1.2 | . . . . 5 ⊢ 𝐵 ∈ Sℋ | |
25 | 23, 24 | shsvai 28351 | . . . 4 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐶 +ℋ 𝑅)) ∧ 𝑦 ∈ 𝐵) → (𝑥 +ℎ 𝑦) ∈ ((𝐴 ∩ (𝐶 +ℋ 𝑅)) +ℋ 𝐵)) |
26 | 23, 24 | shscomi 28350 | . . . 4 ⊢ ((𝐴 ∩ (𝐶 +ℋ 𝑅)) +ℋ 𝐵) = (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))) |
27 | 25, 26 | syl6eleq 2740 | . . 3 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝐶 +ℋ 𝑅)) ∧ 𝑦 ∈ 𝐵) → (𝑥 +ℎ 𝑦) ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅)))) |
28 | 20, 21, 27 | syl2anc 694 | . 2 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → (𝑥 +ℎ 𝑦) ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅)))) |
29 | eleq1 2718 | . . 3 ⊢ (𝑣 = (𝑥 +ℎ 𝑦) → (𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))) ↔ (𝑥 +ℎ 𝑦) ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))))) | |
30 | 29 | ad2antlr 763 | . 2 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → (𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))) ↔ (𝑥 +ℎ 𝑦) ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅))))) |
31 | 28, 30 | mpbird 247 | 1 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ (𝑧 ∈ 𝐶 ∧ (𝑥 −ℎ 𝑧) ∈ 𝑅)) → 𝑣 ∈ (𝐵 +ℋ (𝐴 ∩ (𝐶 +ℋ 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∩ cin 3606 (class class class)co 6690 ℋchil 27904 +ℎ cva 27905 0ℎc0v 27909 −ℎ cmv 27910 Sℋ csh 27913 +ℋ cph 27916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-hilex 27984 ax-hfvadd 27985 ax-hvcom 27986 ax-hvass 27987 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvdistr1 27993 ax-hvdistr2 27994 ax-hvmul0 27995 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-neg 10307 df-grpo 27475 df-ablo 27527 df-hvsub 27956 df-sh 28192 df-shs 28295 |
This theorem is referenced by: 5oalem6 28646 |
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