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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 27286 | . . . 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 1145 | . . 3 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})) |
6 | ovex 7391 | . . . . 5 ⊢ (𝑋 +s 𝑌) ∈ V | |
7 | 6 | snid 4623 | . . . 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 27119 | . . . . . . . . 9 ⊢ ( bday ‘𝑌) ∈ On | |
11 | addspropord.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ No ) | |
12 | oldbday 27233 | . . . . . . . . 9 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑍 ∈ No ) → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌))) | |
13 | 10, 11, 12 | sylancr 588 | . . . . . . . 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 5110 | . . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑌 <s 𝑧 ↔ 𝑌 <s 𝑍)) | |
17 | rightval 27197 | . . . . . . . 8 ⊢ ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑧} | |
18 | 16, 17 | elrab2 3649 | . . . . . . 7 ⊢ (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑍)) |
19 | 14, 15, 18 | sylanbrc 584 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ( R ‘𝑌)) |
20 | eqid 2737 | . . . . . 6 ⊢ (𝑋 +s 𝑍) = (𝑋 +s 𝑍) | |
21 | oveq2 7366 | . . . . . . 7 ⊢ (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍)) | |
22 | 21 | rspceeqv 3596 | . . . . . 6 ⊢ ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)) |
23 | 19, 20, 22 | sylancl 587 | . . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)) |
24 | ovex 7391 | . . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ V | |
25 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑))) | |
26 | 25 | rexbidv 3176 | . . . . . 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 3631 | . . . . 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 4138 | . . . 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 27136 | . 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍)) |
32 | 3, 2 | addscomd 27282 | . 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌)) |
33 | 11, 2 | addscomd 27282 | . 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍)) |
34 | 31, 32, 33 | 3brtr4d 5138 | 1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2714 ∀wral 3065 ∃wrex 3074 ∪ cun 3909 {csn 4587 class class class wbr 5106 Oncon0 6318 ‘cfv 6497 (class class class)co 7358 +no cnadd 8612 No csur 26991 <s cslt 26992 bday cbday 26993 <<s csslt 27123 O cold 27176 L cleft 27178 R cright 27179 +s cadds 27274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-1o 8413 df-2o 8414 df-nadd 8613 df-no 26994 df-slt 26995 df-bday 26996 df-sslt 27124 df-scut 27126 df-0s 27166 df-made 27180 df-old 27181 df-left 27183 df-right 27184 df-norec2 27264 df-adds 27275 |
This theorem is referenced by: addsproplem7 27290 |
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