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Theorem addsproplem1 27450
Description: Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8398 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (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)))))
addsproplem1.2 (𝜑𝐴 No )
addsproplem1.3 (𝜑𝐵 No )
addsproplem1.4 (𝜑𝐶 No )
addsproplem1.5 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
Assertion
Ref Expression
addsproplem1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem addsproplem1
StepHypRef Expression
1 addsproplem1.2 . . 3 (𝜑𝐴 No )
2 addsproplem1.3 . . 3 (𝜑𝐵 No )
3 addsproplem1.4 . . 3 (𝜑𝐶 No )
41, 2, 33jca 1128 . 2 (𝜑 → (𝐴 No 𝐵 No 𝐶 No ))
5 addsproplem.1 . 2 (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)))))
6 addsproplem1.5 . 2 (𝜑 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))))
7 fveq2 6891 . . . . . . 7 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
87oveq1d 7423 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝑦)))
97oveq1d 7423 . . . . . 6 (𝑥 = 𝐴 → (( bday 𝑥) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝑧)))
108, 9uneq12d 4164 . . . . 5 (𝑥 = 𝐴 → ((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
1110eleq1d 2818 . . . 4 (𝑥 = 𝐴 → (((( bday 𝑥) +no ( bday 𝑦)) ∪ (( bday 𝑥) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
12 oveq1 7415 . . . . . 6 (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
1312eleq1d 2818 . . . . 5 (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14 oveq2 7416 . . . . . . 7 (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15 oveq2 7416 . . . . . . 7 (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
1614, 15breq12d 5161 . . . . . 6 (𝑥 = 𝐴 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))
1716imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))
1813, 17anbi12d 631 . . . 4 (𝑥 = 𝐴 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))))
1911, 18imbi12d 344 . . 3 (𝑥 = 𝐴 → ((((( 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)))) ↔ (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))))
20 fveq2 6891 . . . . . . 7 (𝑦 = 𝐵 → ( bday 𝑦) = ( bday 𝐵))
2120oveq2d 7424 . . . . . 6 (𝑦 = 𝐵 → (( bday 𝐴) +no ( bday 𝑦)) = (( bday 𝐴) +no ( bday 𝐵)))
2221uneq1d 4162 . . . . 5 (𝑦 = 𝐵 → ((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))))
2322eleq1d 2818 . . . 4 (𝑦 = 𝐵 → (((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
24 oveq2 7416 . . . . . 6 (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
2524eleq1d 2818 . . . . 5 (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26 breq1 5151 . . . . . 6 (𝑦 = 𝐵 → (𝑦 <s 𝑧𝐵 <s 𝑧))
27 oveq1 7415 . . . . . . 7 (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
2827breq1d 5158 . . . . . 6 (𝑦 = 𝐵 → ((𝑦 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))
2926, 28imbi12d 344 . . . . 5 (𝑦 = 𝐵 → ((𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))
3025, 29anbi12d 631 . . . 4 (𝑦 = 𝐵 → (((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))))
3123, 30imbi12d 344 . . 3 (𝑦 = 𝐵 → ((((( bday 𝐴) +no ( bday 𝑦)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))))
32 fveq2 6891 . . . . . . 7 (𝑧 = 𝐶 → ( bday 𝑧) = ( bday 𝐶))
3332oveq2d 7424 . . . . . 6 (𝑧 = 𝐶 → (( bday 𝐴) +no ( bday 𝑧)) = (( bday 𝐴) +no ( bday 𝐶)))
3433uneq2d 4163 . . . . 5 (𝑧 = 𝐶 → ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) = ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))))
3534eleq1d 2818 . . . 4 (𝑧 = 𝐶 → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) ↔ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍)))))
36 breq2 5152 . . . . . 6 (𝑧 = 𝐶 → (𝐵 <s 𝑧𝐵 <s 𝐶))
37 oveq1 7415 . . . . . . 7 (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
3837breq2d 5160 . . . . . 6 (𝑧 = 𝐶 → ((𝐵 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))
3936, 38imbi12d 344 . . . . 5 (𝑧 = 𝐶 → ((𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
4039anbi2d 629 . . . 4 (𝑧 = 𝐶 → (((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))))
4135, 40imbi12d 344 . . 3 (𝑧 = 𝐶 → ((((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝑧))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
4219, 31, 41rspc3v 3627 . 2 ((𝐴 No 𝐵 No 𝐶 No ) → (∀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)))) → (((( bday 𝐴) +no ( bday 𝐵)) ∪ (( bday 𝐴) +no ( bday 𝐶))) ∈ ((( bday 𝑋) +no ( bday 𝑌)) ∪ (( bday 𝑋) +no ( bday 𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
434, 5, 6, 42syl3c 66 1 (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  cun 3946   class class class wbr 5148  cfv 6543  (class class class)co 7408   +no cnadd 8663   No csur 27140   <s cslt 27141   bday cbday 27142   +s cadds 27440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411
This theorem is referenced by:  addsproplem2  27451  addsproplem6  27455
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