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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddssat | Structured version Visualization version GIF version |
Description: A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.) |
Ref | Expression |
---|---|
padd0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
padd0.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddssat | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | eqid 2821 | . . 3 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | padd0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | padd0.p | . . 3 ⊢ + = (+𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | paddval 36949 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})) |
6 | unss 4160 | . . . . . 6 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ↔ (𝑋 ∪ 𝑌) ⊆ 𝐴) | |
7 | 6 | biimpi 218 | . . . . 5 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ 𝐴) |
8 | ssrab2 4056 | . . . . 5 ⊢ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)} ⊆ 𝐴 | |
9 | 7, 8 | jctir 523 | . . . 4 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ∪ 𝑌) ⊆ 𝐴 ∧ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)} ⊆ 𝐴)) |
10 | unss 4160 | . . . 4 ⊢ (((𝑋 ∪ 𝑌) ⊆ 𝐴 ∧ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)} ⊆ 𝐴) ↔ ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) ⊆ 𝐴) | |
11 | 9, 10 | sylib 220 | . . 3 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) ⊆ 𝐴) |
12 | 11 | 3adant1 1126 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) ⊆ 𝐴) |
13 | 5, 12 | eqsstrd 4005 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 {crab 3142 ∪ cun 3934 ⊆ wss 3936 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 lecple 16572 joincjn 17554 Atomscatm 36414 +𝑃cpadd 36946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-padd 36947 |
This theorem is referenced by: paddasslem8 36978 paddasslem11 36981 paddasslem12 36982 paddasslem13 36983 paddasslem16 36986 paddasslem17 36987 paddass 36989 padd4N 36991 paddclN 36993 pmodl42N 37002 pclunN 37049 paddunN 37078 pmapocjN 37081 pclfinclN 37101 osumcllem1N 37107 osumcllem2N 37108 osumcllem9N 37115 osumcllem11N 37117 osumclN 37118 pexmidlem6N 37126 pexmidlem8N 37128 pl42lem3N 37132 |
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