Theorem addsproplem7 27905

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 27722	. . . 4 ⊢ ( bday ‘𝑌) ∈ On
2		fvex 6903	. . . . 5 ⊢ ( bday ‘𝑌) ∈ V
3	2	elon 6374	. . . 4 ⊢ (( bday ‘𝑌) ∈ On ↔ Ord ( bday ‘𝑌))
4	1, 3	mpbi 229	. . 3 ⊢ Ord ( bday ‘𝑌)
5		bdayelon 27722	. . . 4 ⊢ ( bday ‘𝑍) ∈ On
6		fvex 6903	. . . . 5 ⊢ ( bday ‘𝑍) ∈ V
7	6	elon 6374	. . . 4 ⊢ (( bday ‘𝑍) ∈ On ↔ Ord ( bday ‘𝑍))
8	5, 7	mpbi 229	. . 3 ⊢ Ord ( bday ‘𝑍)
9		ordtri3or 6397	. . 3 ⊢ ((Ord ( bday ‘𝑌) ∧ Ord ( bday ‘𝑍)) → (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
10	4, 8, 9	mp2an 690	. 2 ⊢ (( bday ‘𝑌) ∈ ( bday ‘𝑍) ∨ ( bday ‘𝑌) = ( bday ‘𝑍) ∨ ( bday ‘𝑍) ∈ ( bday ‘𝑌))
11		simpl 481	. . . . . 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 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → ( bday ‘𝑌) ∈ ( bday ‘𝑍))
23	13, 15, 17, 19, 21, 22	addsproplem4 27902	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) ∈ ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
24	23	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) ∈ ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
25		simpl 481	. . . . . 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 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → ( bday ‘𝑌) = ( bday ‘𝑍))
32	26, 27, 28, 29, 30, 31	addsproplem6 27904	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑌) = ( bday ‘𝑍)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
33	32	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝑌) = ( bday ‘𝑍) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
34	12	adantr 479	. . . . 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 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑋 ∈ No )
36	16	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 ∈ No )
37	18	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑍 ∈ No )
38	20	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → 𝑌 <s 𝑍)
39		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → ( bday ‘𝑍) ∈ ( bday ‘𝑌))
40	34, 35, 36, 37, 38, 39	addsproplem5 27903	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝑍) ∈ ( bday ‘𝑌)) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))
41	40	ex 411	. . 3 ⊢ (𝜑 → (( bday ‘𝑍) ∈ ( bday ‘𝑌) → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
42	24, 33, 41	3jaod 1425	. 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 394   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  ∀wral 3051   ∪ cun 3939   class class class wbr 5144  Ord word 6364  Oncon0 6365  ‘cfv 6543  (class class class)co 7413   +no cnadd 8679   No csur 27586   <s cslt 27587   bday cbday 27588   +_s cadds 27889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-nadd 8680  df-no 27589  df-slt 27590  df-bday 27591  df-sslt 27727  df-scut 27729  df-0s 27770  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec2 27879  df-adds 27890

This theorem is referenced by:  addsprop  27906