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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddvaln0N | Structured version Visualization version GIF version |
Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
paddfval.l | ⊢ ≤ = (le‘𝐾) |
paddfval.j | ⊢ ∨ = (join‘𝐾) |
paddfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
paddfval.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
paddvaln0N | ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | paddfval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | paddfval.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
3 | paddfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | paddfval.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
5 | 1, 2, 3, 4 | elpaddn0 37578 | . . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑠 ∈ (𝑋 + 𝑌) ↔ (𝑠 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑠 ≤ (𝑞 ∨ 𝑟)))) |
6 | breq1 5071 | . . . . 5 ⊢ (𝑝 = 𝑠 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑠 ≤ (𝑞 ∨ 𝑟))) | |
7 | 6 | 2rexbidv 3227 | . . . 4 ⊢ (𝑝 = 𝑠 → (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟) ↔ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑠 ≤ (𝑞 ∨ 𝑟))) |
8 | 7 | elrab 3615 | . . 3 ⊢ (𝑠 ∈ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ (𝑠 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑠 ≤ (𝑞 ∨ 𝑟))) |
9 | 5, 8 | bitr4di 292 | . 2 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑠 ∈ (𝑋 + 𝑌) ↔ 𝑠 ∈ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)})) |
10 | 9 | eqrdv 2736 | 1 ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 ≠ wne 2941 ∃wrex 3063 {crab 3066 ⊆ wss 3881 ∅c0 4252 class class class wbr 5068 ‘cfv 6398 (class class class)co 7232 lecple 16834 joincjn 17843 Latclat 17962 Atomscatm 37041 +𝑃cpadd 37573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-1st 7780 df-2nd 7781 df-lub 17877 df-join 17879 df-lat 17963 df-ats 37045 df-padd 37574 |
This theorem is referenced by: (None) |
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