Theorem addsproplem7 28023

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	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <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

Step	Hyp	Ref	Expression
1		bdayelon 27836	. . . 4 ⊢ ( bday ‘𝑌) ∈ On
2		fvex 6920	. . . . 5 ⊢ ( bday ‘𝑌) ∈ V
3	2	elon 6395	. . . 4 ⊢ (( bday ‘𝑌) ∈ On ↔ Ord ( bday ‘𝑌))
4	1, 3	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑌)
5		bdayelon 27836	. . . 4 ⊢ ( bday ‘𝑍) ∈ On
6		fvex 6920	. . . . 5 ⊢ ( bday ‘𝑍) ∈ V
7	6	elon 6395	. . . 4 ⊢ (( bday ‘𝑍) ∈ On ↔ Ord ( bday ‘𝑍))
8	5, 7	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑍)
9		ordtri3or 6418	. . 3 ⊢ ((Ord ( bday ‘𝑌) ∧ Ord ( bday ‘𝑍)) → (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
10	4, 8, 9	mp2an 692	. 2 ⊢ (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌))
11		simpl 482	. . . . . 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)))))
13	11, 12	syl 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 ⊢ (𝜑 → 𝑋 ∈ No )
15	11, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑋 ∈ No )
16		addspropord.3	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ No )
17	11, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 ∈ No )
18		addspropord.4	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ No )
19	11, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑍 ∈ No )
20		addspropord.5	. . . . . 6 ⊢ (𝜑 → 𝑌 <s 𝑍)
21	11, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → 𝑌 <s 𝑍)
22		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → ( bday ‘𝑌) ∈ ( bday ‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 28020	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) ∈ ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 482	. . . . . 6 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝜑)
26	25, 12	syl 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)))))
27	25, 14	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑋 ∈ No )
28	25, 16	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 ∈ No )
29	25, 18	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑍 ∈ No )
30	25, 20	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → 𝑌 <s 𝑍)
31		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → ( bday ‘𝑌) = ( bday ‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 28022	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) = ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 480	. . . . 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)))))
35	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑋 ∈ No )
36	16	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 ∈ No )
37	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑍 ∈ No )
38	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → ( bday ‘𝑍) ∈ ( bday ‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 28021	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 412	. . 3 ⊢ (𝜑 → (( bday ‘𝑍) ∈ ( bday ‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1428	. 2 ⊢ (𝜑 → ((( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
43	10, 42	mpi 20	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∪ cun 3961   class class class wbr 5148  Ord word 6385  Oncon0 6386  ‘cfv 6563  (class class class)co 7431   +no cnadd 8702   No csur 27699   <s cslt 27700   bday cbday 27701   +_s cadds 28007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008

This theorem is referenced by:  addsprop  28024