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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspadd2 | Structured version Visualization version GIF version |
Description: A projective subspace sum is a superset of its second summand. (ssun2 3739 analog.) (Contributed by NM, 3-Jan-2012.) |
Ref | Expression |
---|---|
padd0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
padd0.p | ⊢ + = (+𝑃‘𝐾) |
Ref | Expression |
---|---|
sspadd2 | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3739 | . . 3 ⊢ 𝑋 ⊆ (𝑌 ∪ 𝑋) | |
2 | ssun1 3738 | . . 3 ⊢ (𝑌 ∪ 𝑋) ⊆ ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) | |
3 | 1, 2 | sstri 3577 | . 2 ⊢ 𝑋 ⊆ ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) |
4 | eqid 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2610 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
6 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
7 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
8 | 4, 5, 6, 7 | paddval 34102 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → (𝑌 + 𝑋) = ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})) |
9 | 8 | 3com23 1263 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑌 + 𝑋) = ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)})) |
10 | 3, 9 | syl5sseqr 3617 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 ∪ cun 3538 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 lecple 15775 joincjn 16767 Atomscatm 33568 +𝑃cpadd 34099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-padd 34100 |
This theorem is referenced by: paddasslem11 34134 paddasslem12 34135 paddssw2 34148 pmodlem2 34151 pmodl42N 34155 osumcllem10N 34269 pexmidlem7N 34280 pl42lem3N 34285 |
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