Theorem addsproplem7 27919

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 27716	. . . 4 ⊢ ( bday ‘𝑌) ∈ On
2		fvex 6835	. . . . 5 ⊢ ( bday ‘𝑌) ∈ V
3	2	elon 6315	. . . 4 ⊢ (( bday ‘𝑌) ∈ On ↔ Ord ( bday ‘𝑌))
4	1, 3	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑌)
5		bdayelon 27716	. . . 4 ⊢ ( bday ‘𝑍) ∈ On
6		fvex 6835	. . . . 5 ⊢ ( bday ‘𝑍) ∈ V
7	6	elon 6315	. . . 4 ⊢ (( bday ‘𝑍) ∈ On ↔ Ord ( bday ‘𝑍))
8	5, 7	mpbi 230	. . 3 ⊢ Ord ( bday ‘𝑍)
9		ordtri3or 6338	. . 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 27916	. . . 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 27918	. . . 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 27917	. . . 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 1431	. 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 1541   ∈ wcel 2111  ∀wral 3047   ∪ cun 3900   class class class wbr 5091  Ord word 6305  Oncon0 6306  ‘cfv 6481  (class class class)co 7346   +no cnadd 8580   No csur 27579   <s cslt 27580   bday cbday 27581   +_s cadds 27903

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27582  df-slt 27583  df-bday 27584  df-sslt 27722  df-scut 27724  df-0s 27769  df-made 27789  df-old 27790  df-left 27792  df-right 27793  df-norec2 27893  df-adds 27904

This theorem is referenced by:  addsprop  27920