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Theorem addsproplem7 27290
Description: Lemma for surreal addition properties. Putting together the three previous lemmas, we now show the second half of the inductive hypothesis unconditionally. (Contributed by Scott Fenton, 21-Jan-2025.)
Hypotheses
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 (𝜑𝑋 No )
addspropord.3 (𝜑𝑌 No )
addspropord.4 (𝜑𝑍 No )
addspropord.5 (𝜑𝑌 <s 𝑍)
Assertion
Ref Expression
addsproplem7 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem addsproplem7
StepHypRef Expression
1 bdayelon 27119 . . . 4 ( bday 𝑌) ∈ On
2 fvex 6856 . . . . 5 ( bday 𝑌) ∈ V
32elon 6327 . . . 4 (( bday 𝑌) ∈ On ↔ Ord ( bday 𝑌))
41, 3mpbi 229 . . 3 Ord ( bday 𝑌)
5 bdayelon 27119 . . . 4 ( bday 𝑍) ∈ On
6 fvex 6856 . . . . 5 ( bday 𝑍) ∈ V
76elon 6327 . . . 4 (( bday 𝑍) ∈ On ↔ Ord ( bday 𝑍))
85, 7mpbi 229 . . 3 Ord ( bday 𝑍)
9 ordtri3or 6350 . . 3 ((Ord ( bday 𝑌) ∧ Ord ( bday 𝑍)) → (( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌)))
104, 8, 9mp2an 691 . 2 (( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌))
11 simpl 484 . . . . . 6 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝜑)
12 addsproplem.1 . . . . . 6 (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)))))
1311, 12syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → ∀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)))))
14 addspropord.2 . . . . . 6 (𝜑𝑋 No )
1511, 14syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑋 No )
16 addspropord.3 . . . . . 6 (𝜑𝑌 No )
1711, 16syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑌 No )
18 addspropord.4 . . . . . 6 (𝜑𝑍 No )
1911, 18syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑍 No )
20 addspropord.5 . . . . . 6 (𝜑𝑌 <s 𝑍)
2111, 20syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → 𝑌 <s 𝑍)
22 simpr 486 . . . . 5 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → ( bday 𝑌) ∈ ( bday 𝑍))
2313, 15, 17, 19, 21, 22addsproplem4 27287 . . . 4 ((𝜑 ∧ ( bday 𝑌) ∈ ( bday 𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
2423ex 414 . . 3 (𝜑 → (( bday 𝑌) ∈ ( bday 𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25 simpl 484 . . . . . 6 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝜑)
2625, 12syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → ∀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)))))
2725, 14syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑋 No )
2825, 16syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑌 No )
2925, 18syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑍 No )
3025, 20syl 17 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → 𝑌 <s 𝑍)
31 simpr 486 . . . . 5 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → ( bday 𝑌) = ( bday 𝑍))
3226, 27, 28, 29, 30, 31addsproplem6 27289 . . . 4 ((𝜑 ∧ ( bday 𝑌) = ( bday 𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
3332ex 414 . . 3 (𝜑 → (( bday 𝑌) = ( bday 𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
3412adantr 482 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → ∀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)))))
3514adantr 482 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑋 No )
3616adantr 482 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑌 No )
3718adantr 482 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑍 No )
3820adantr 482 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → 𝑌 <s 𝑍)
39 simpr 486 . . . . 5 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → ( bday 𝑍) ∈ ( bday 𝑌))
4034, 35, 36, 37, 38, 39addsproplem5 27288 . . . 4 ((𝜑 ∧ ( bday 𝑍) ∈ ( bday 𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
4140ex 414 . . 3 (𝜑 → (( bday 𝑍) ∈ ( bday 𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
4224, 33, 413jaod 1429 . 2 (𝜑 → ((( bday 𝑌) ∈ ( bday 𝑍) ∨ ( bday 𝑌) = ( bday 𝑍) ∨ ( bday 𝑍) ∈ ( bday 𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
4310, 42mpi 20 1 (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3o 1087   = wceq 1542  wcel 2107  wral 3065  cun 3909   class class class wbr 5106  Ord word 6317  Oncon0 6318  cfv 6497  (class class class)co 7358   +no cnadd 8612   No csur 26991   <s cslt 26992   bday cbday 26993   +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:  addsprop  27291
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