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Mirrors > Home > HSE Home > Th. List > 3oalem1 | Structured version Visualization version GIF version |
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
3oalem1.1 | ⊢ 𝐵 ∈ Cℋ |
3oalem1.2 | ⊢ 𝐶 ∈ Cℋ |
3oalem1.3 | ⊢ 𝑅 ∈ Cℋ |
3oalem1.4 | ⊢ 𝑆 ∈ Cℋ |
Ref | Expression |
---|---|
3oalem1 | ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3oalem1.1 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
2 | 1 | cheli 30223 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ ℋ) |
3 | 3oalem1.3 | . . . . 5 ⊢ 𝑅 ∈ Cℋ | |
4 | 3 | cheli 30223 | . . . 4 ⊢ (𝑦 ∈ 𝑅 → 𝑦 ∈ ℋ) |
5 | 2, 4 | anim12i 614 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) |
6 | hvaddcl 30003 | . . . . 5 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 +ℎ 𝑦) ∈ ℋ) | |
7 | eleq1 2822 | . . . . 5 ⊢ (𝑣 = (𝑥 +ℎ 𝑦) → (𝑣 ∈ ℋ ↔ (𝑥 +ℎ 𝑦) ∈ ℋ)) | |
8 | 6, 7 | syl5ibrcom 247 | . . . 4 ⊢ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑣 = (𝑥 +ℎ 𝑦) → 𝑣 ∈ ℋ)) |
9 | 8 | imdistani 570 | . . 3 ⊢ (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ)) |
10 | 5, 9 | sylan 581 | . 2 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ)) |
11 | 3oalem1.2 | . . . . 5 ⊢ 𝐶 ∈ Cℋ | |
12 | 11 | cheli 30223 | . . . 4 ⊢ (𝑧 ∈ 𝐶 → 𝑧 ∈ ℋ) |
13 | 3oalem1.4 | . . . . 5 ⊢ 𝑆 ∈ Cℋ | |
14 | 13 | cheli 30223 | . . . 4 ⊢ (𝑤 ∈ 𝑆 → 𝑤 ∈ ℋ) |
15 | 12, 14 | anim12i 614 | . . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) |
16 | 15 | adantr 482 | . 2 ⊢ (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)) → (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ)) |
17 | 10, 16 | anim12i 614 | 1 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑣 ∈ ℋ) ∧ (𝑧 ∈ ℋ ∧ 𝑤 ∈ ℋ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7361 ℋchba 29910 +ℎ cva 29911 Cℋ cch 29920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-hilex 29990 ax-hfvadd 29991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-sh 30198 df-ch 30212 |
This theorem is referenced by: 3oalem2 30654 |
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