Theorem addsproplem6 27991

Description: Lemma for surreal addition properties. Finally, we show the second half of the induction hypothesis when 𝑌 and 𝑍 are the same age. (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)
addsproplem6.6	⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))

Assertion

Ref	Expression
addsproplem6	⊢ (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 addsproplem6

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addspropord.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ No )
2		addspropord.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ No )
3		addsproplem6.6	. . . 4 ⊢ (𝜑 → ( bday ‘𝑌) = ( bday ‘𝑍))
4		addspropord.5	. . . 4 ⊢ (𝜑 → 𝑌 <s 𝑍)
5		nodense 27681	. . . 4 ⊢ (((𝑌 ∈ No ∧ 𝑍 ∈ No ) ∧ (( bday ‘𝑌) = ( bday ‘𝑍) ∧ 𝑌 <s 𝑍)) → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
6	1, 2, 3, 4, 5	syl22anc 844	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ No (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))
7		addsproplem.1	. . . . . . 7 ⊢ (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)))))
8		addspropord.2	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ No )
9	7, 8, 1	addsproplem3 27988	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
10	9	simp1d 1148	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ No )
11	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ No )
12	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 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)))))
13	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑋 ∈ No )
14		simprl 776	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ No )
15		unidm 4094	. . . . . . 7 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) = (( bday ‘𝑋) +no ( bday ‘𝑚))
16		simprr1 1228	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑌))
17		bdayon 27769	. . . . . . . . . 10 ⊢ ( bday ‘𝑚) ∈ On
18		bdayon 27769	. . . . . . . . . 10 ⊢ ( bday ‘𝑌) ∈ On
19		bdayon 27769	. . . . . . . . . 10 ⊢ ( bday ‘𝑋) ∈ On
20		naddel2 8621	. . . . . . . . . 10 ⊢ ((( bday ‘𝑚) ∈ On ∧ ( bday ‘𝑌) ∈ On ∧ ( bday ‘𝑋) ∈ On) → (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌))))
21	17, 18, 19, 20	mp3an 1469	. . . . . . . . 9 ⊢ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ↔ (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
22	16, 21	sylib 219	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)))
23		elun1 4118	. . . . . . . 8 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ (( bday ‘𝑋) +no ( bday ‘𝑌)) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
24	22, 23	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (( bday ‘𝑋) +no ( bday ‘𝑚)) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
25	15, 24	eqeltrid 2844	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((( bday ‘𝑋) +no ( bday ‘𝑚)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑚))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))))
26	12, 13, 14, 14, 25	addsproplem1 27986	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑚) ∈ No ∧ (𝑚 <s 𝑚 → (𝑚 +s 𝑋) <s (𝑚 +s 𝑋))))
27	26	simpld 495	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ No )
28		uncom 4095	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
29	28	eleq2i 2832	. . . . . . . . . . 11 ⊢ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ↔ ((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))))
30	29	imbi1i 350	. . . . . . . . . 10 ⊢ ((((( 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 ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
31	30	ralbii 3086	. . . . . . . . 9 ⊢ (∀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)))) ↔ ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
32	31	2ralbii 3115	. . . . . . . 8 ⊢ (∀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)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
33	7, 32	sylib 219	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
34	33, 8, 2	addsproplem3 27988	. . . . . 6 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
35	34	simp1d 1148	. . . . 5 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ No )
36	35	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ No )
37	9	simp3d 1150	. . . . . 6 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
39		ovex 7396	. . . . . . 7 ⊢ (𝑋 +s 𝑌) ∈ V
40	39	snid 4601	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
41	40	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
42		oldbday 27918	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
43	18, 14, 42	sylancr 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑌)))
44	16, 43	mpbird 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑌)))
45		simprr2 1229	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑌 <s 𝑚)
46		elright 27869	. . . . . . . . 9 ⊢ (𝑚 ∈ ( R ‘𝑌) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑚))
47	44, 45, 46	sylanbrc 589	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( R ‘𝑌))
48		eqid 2740	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)
49		oveq2 7371	. . . . . . . . 9 ⊢ (ℎ = 𝑚 → (𝑋 +s ℎ) = (𝑋 +s 𝑚))
50	49	rspceeqv 3590	. . . . . . . 8 ⊢ ((𝑚 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
51	47, 48, 50	sylancl 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
52		ovex 7396	. . . . . . . 8 ⊢ (𝑋 +s 𝑚) ∈ V
53		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑔 = (𝑋 +s 𝑚) → (𝑔 = (𝑋 +s ℎ) ↔ (𝑋 +s 𝑚) = (𝑋 +s ℎ)))
54	53	rexbidv 3164	. . . . . . . 8 ⊢ (𝑔 = (𝑋 +s 𝑚) → (∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ)))
55	52, 54	elab 3624	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} ↔ ∃ℎ ∈ ( R ‘𝑌)(𝑋 +s 𝑚) = (𝑋 +s ℎ))
56	51, 55	sylibr 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})
57		elun2 4119	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)} → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
58	56, 57	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
59	38, 41, 58	sltssepcd 27789	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑚))
60	33	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
61	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑍 ∈ No )
62	60, 13, 61	addsproplem3 27988	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑍)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑔 = (𝑋 +s ℎ)})))
63	62	simp2d 1149	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
64	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑌) = ( bday ‘𝑍))
65	16, 64	eleqtrd 2842	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ( bday ‘𝑚) ∈ ( bday ‘𝑍))
66		bdayon 27769	. . . . . . . . . . 11 ⊢ ( bday ‘𝑍) ∈ On
67		oldbday 27918	. . . . . . . . . . 11 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑚 ∈ No ) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
68	66, 14, 67	sylancr 593	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑚 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑚) ∈ ( bday ‘𝑍)))
69	65, 68	mpbird 258	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( O ‘( bday ‘𝑍)))
70		simprr3 1230	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 <s 𝑍)
71		elleft 27868	. . . . . . . . 9 ⊢ (𝑚 ∈ ( L ‘𝑍) ↔ (𝑚 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑚 <s 𝑍))
72	69, 70, 71	sylanbrc 589	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → 𝑚 ∈ ( L ‘𝑍))
73		oveq2 7371	. . . . . . . . 9 ⊢ (𝑑 = 𝑚 → (𝑋 +s 𝑑) = (𝑋 +s 𝑚))
74	73	rspceeqv 3590	. . . . . . . 8 ⊢ ((𝑚 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑚) = (𝑋 +s 𝑚)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
75	72, 48, 74	sylancl 592	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
76		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑐 = (𝑋 +s 𝑚) → (𝑐 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
77	76	rexbidv 3164	. . . . . . . 8 ⊢ (𝑐 = (𝑋 +s 𝑚) → (∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑)))
78	52, 77	elab 3624	. . . . . . 7 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑚) = (𝑋 +s 𝑑))
79	75, 78	sylibr 235	. . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)})
80		elun2 4119	. . . . . 6 ⊢ ((𝑋 +s 𝑚) ∈ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
81	79, 80	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) ∈ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑍)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑐 = (𝑋 +s 𝑑)}))
82		ovex 7396	. . . . . . 7 ⊢ (𝑋 +s 𝑍) ∈ V
83	82	snid 4601	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
84	83	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
85	63, 81, 84	sltssepcd 27789	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑚) <s (𝑋 +s 𝑍))
86	11, 27, 36, 59, 85	ltstrd 27752	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ No ∧ (( bday ‘𝑚) ∈ ( bday ‘𝑌) ∧ 𝑌 <s 𝑚 ∧ 𝑚 <s 𝑍))) → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
87	6, 86	rexlimddv 3147	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
88	1, 8	addscomd 27984	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
89	2, 8	addscomd 27984	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
90	87, 88, 89	3brtr4d 5111	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064   ∪ cun 3888  {csn 4562   class class class wbr 5079  Oncon0 6317  ‘cfv 6492  (class class class)co 7363   +no cnadd 8598   No csur 27628   <s clts 27629   bday cbday 27630   <<s cslts 27774   O cold 27840   L cleft 27842   R cright 27843   +_s cadds 27976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec2 27966  df-adds 27977

This theorem is referenced by:  addsproplem7  27992