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Mirrors > Home > HSE Home > Th. List > 3oalem3 | 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 |
---|---|
3oalem3 | ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ⊆ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3oalem1.1 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
2 | 3oalem1.3 | . . . . . . 7 ⊢ 𝑅 ∈ Cℋ | |
3 | 1, 2 | chseli 29242 | . . . . . 6 ⊢ (𝑣 ∈ (𝐵 +ℋ 𝑅) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝑅 𝑣 = (𝑥 +ℎ 𝑦)) |
4 | r2ex 3262 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝑅 𝑣 = (𝑥 +ℎ 𝑦) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦))) | |
5 | 3, 4 | bitri 278 | . . . . 5 ⊢ (𝑣 ∈ (𝐵 +ℋ 𝑅) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦))) |
6 | 3oalem1.2 | . . . . . . 7 ⊢ 𝐶 ∈ Cℋ | |
7 | 3oalem1.4 | . . . . . . 7 ⊢ 𝑆 ∈ Cℋ | |
8 | 6, 7 | chseli 29242 | . . . . . 6 ⊢ (𝑣 ∈ (𝐶 +ℋ 𝑆) ↔ ∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝑆 𝑣 = (𝑧 +ℎ 𝑤)) |
9 | r2ex 3262 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐶 ∃𝑤 ∈ 𝑆 𝑣 = (𝑧 +ℎ 𝑤) ↔ ∃𝑧∃𝑤((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) | |
10 | 8, 9 | bitri 278 | . . . . 5 ⊢ (𝑣 ∈ (𝐶 +ℋ 𝑆) ↔ ∃𝑧∃𝑤((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) |
11 | 5, 10 | anbi12i 629 | . . . 4 ⊢ ((𝑣 ∈ (𝐵 +ℋ 𝑅) ∧ 𝑣 ∈ (𝐶 +ℋ 𝑆)) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ∃𝑧∃𝑤((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)))) |
12 | elin 3897 | . . . 4 ⊢ (𝑣 ∈ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ↔ (𝑣 ∈ (𝐵 +ℋ 𝑅) ∧ 𝑣 ∈ (𝐶 +ℋ 𝑆))) | |
13 | 4exdistrv 1957 | . . . 4 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) ↔ (∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ∃𝑧∃𝑤((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)))) | |
14 | 11, 12, 13 | 3bitr4i 306 | . . 3 ⊢ (𝑣 ∈ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ↔ ∃𝑥∃𝑧∃𝑦∃𝑤(((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤)))) |
15 | 1, 6, 2, 7 | 3oalem2 29446 | . . . . 5 ⊢ ((((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))) |
16 | 15 | exlimivv 1933 | . . . 4 ⊢ (∃𝑦∃𝑤(((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))) |
17 | 16 | exlimivv 1933 | . . 3 ⊢ (∃𝑥∃𝑧∃𝑦∃𝑤(((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝑅) ∧ 𝑣 = (𝑥 +ℎ 𝑦)) ∧ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 = (𝑧 +ℎ 𝑤))) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))) |
18 | 14, 17 | sylbi 220 | . 2 ⊢ (𝑣 ∈ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) → 𝑣 ∈ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆)))))) |
19 | 18 | ssriv 3919 | 1 ⊢ ((𝐵 +ℋ 𝑅) ∩ (𝐶 +ℋ 𝑆)) ⊆ (𝐵 +ℋ (𝑅 ∩ (𝑆 +ℋ ((𝐵 +ℋ 𝐶) ∩ (𝑅 +ℋ 𝑆))))) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 (class class class)co 7135 +ℎ cva 28703 Cℋ cch 28712 +ℋ cph 28714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hilex 28782 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvdistr1 28791 ax-hvdistr2 28792 ax-hvmul0 28793 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-nn 11626 df-grpo 28276 df-ablo 28328 df-hvsub 28754 df-hlim 28755 df-sh 28990 df-ch 29004 df-shs 29091 |
This theorem is referenced by: 3oai 29451 |
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