| Mathbox for Norm Megill |
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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 4179 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 4179 | . . 3 ⊢ 𝑋 ⊆ (𝑌 ∪ 𝑋) | |
| 2 | ssun1 4178 | . . 3 ⊢ (𝑌 ∪ 𝑋) ⊆ ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) | |
| 3 | 1, 2 | sstri 3993 | . 2 ⊢ 𝑋 ⊆ ((𝑌 ∪ 𝑋) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑌 ∃𝑟 ∈ 𝑋 𝑝(le‘𝐾)(𝑞(join‘𝐾)𝑟)}) |
| 4 | eqid 2737 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 5 | eqid 2737 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
| 6 | padd0.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 7 | padd0.p | . . . 4 ⊢ + = (+𝑃‘𝐾) | |
| 8 | 4, 5, 6, 7 | paddval 39800 | . . 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 1127 | . 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 | sseqtrrid 4027 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 ∪ cun 3949 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 lecple 17304 joincjn 18357 Atomscatm 39264 +𝑃cpadd 39797 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-padd 39798 |
| This theorem is referenced by: paddasslem11 39832 paddasslem12 39833 paddssw2 39846 pmodlem2 39849 pmodl42N 39853 osumcllem10N 39967 pexmidlem7N 39978 pl42lem3N 39983 |
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