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Mirrors > Home > MPE Home > Th. List > addsproplem5 | Structured version Visualization version GIF version |
Description: Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.) |
Ref | Expression |
---|---|
addsproplem.1 | ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) |
addspropord.2 | ⊢ (𝜑 → 𝑋 ∈ No ) |
addspropord.3 | ⊢ (𝜑 → 𝑌 ∈ No ) |
addspropord.4 | ⊢ (𝜑 → 𝑍 ∈ No ) |
addspropord.5 | ⊢ (𝜑 → 𝑌 <s 𝑍) |
addsproplem5.6 | ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌)) |
Ref | Expression |
---|---|
addsproplem5 | ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addsproplem.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))) | |
2 | addspropord.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ No ) | |
3 | addspropord.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ No ) | |
4 | 1, 2, 3 | addsproplem3 27708 | . . . 4 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( L ‘𝑌)𝑔 = (𝑋 +s ℎ)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))) |
5 | 4 | simp3d 1143 | . . 3 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})) |
6 | ovex 7445 | . . . . 5 ⊢ (𝑋 +s 𝑌) ∈ V | |
7 | 6 | snid 4664 | . . . 4 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)} |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}) |
9 | addsproplem5.6 | . . . . . . . 8 ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌)) | |
10 | bdayelon 27529 | . . . . . . . . 9 ⊢ ( bday ‘𝑌) ∈ On | |
11 | addspropord.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ No ) | |
12 | oldbday 27647 | . . . . . . . . 9 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑍 ∈ No ) → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌))) | |
13 | 10, 11, 12 | sylancr 586 | . . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌))) |
14 | 9, 13 | mpbird 257 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ( O ‘( bday ‘𝑌))) |
15 | addspropord.5 | . . . . . . 7 ⊢ (𝜑 → 𝑌 <s 𝑍) | |
16 | breq2 5152 | . . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑌 <s 𝑧 ↔ 𝑌 <s 𝑍)) | |
17 | rightval 27611 | . . . . . . . 8 ⊢ ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑧} | |
18 | 16, 17 | elrab2 3686 | . . . . . . 7 ⊢ (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑍)) |
19 | 14, 15, 18 | sylanbrc 582 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ( R ‘𝑌)) |
20 | eqid 2731 | . . . . . 6 ⊢ (𝑋 +s 𝑍) = (𝑋 +s 𝑍) | |
21 | oveq2 7420 | . . . . . . 7 ⊢ (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍)) | |
22 | 21 | rspceeqv 3633 | . . . . . 6 ⊢ ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)) |
23 | 19, 20, 22 | sylancl 585 | . . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)) |
24 | ovex 7445 | . . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ V | |
25 | eqeq1 2735 | . . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑))) | |
26 | 25 | rexbidv 3177 | . . . . . 6 ⊢ (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))) |
27 | 24, 26 | elab 3668 | . . . . 5 ⊢ ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)) |
28 | 23, 27 | sylibr 233 | . . . 4 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}) |
29 | elun2 4177 | . . . 4 ⊢ ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})) | |
30 | 28, 29 | syl 17 | . . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})) |
31 | 5, 8, 30 | ssltsepcd 27547 | . 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍)) |
32 | 3, 2 | addscomd 27704 | . 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌)) |
33 | 11, 2 | addscomd 27704 | . 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍)) |
34 | 31, 32, 33 | 3brtr4d 5180 | 1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 {cab 2708 ∀wral 3060 ∃wrex 3069 ∪ cun 3946 {csn 4628 class class class wbr 5148 Oncon0 6364 ‘cfv 6543 (class class class)co 7412 +no cnadd 8670 No csur 27394 <s cslt 27395 bday cbday 27396 <<s csslt 27533 O cold 27590 L cleft 27592 R cright 27593 +s cadds 27696 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-1o 8472 df-2o 8473 df-nadd 8671 df-no 27397 df-slt 27398 df-bday 27399 df-sslt 27534 df-scut 27536 df-0s 27577 df-made 27594 df-old 27595 df-left 27597 df-right 27598 df-norec2 27686 df-adds 27697 |
This theorem is referenced by: addsproplem7 27712 |
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